Involutions of Azumaya algebras

Uriya A. First Uriya A. First
Department of Mathematics
University of Haifa
Haifa 31905
Israel
uriya.first@gmail.com
 and  Ben Williams Ben Williams
Department of Mathematics
University of British Columbia
Vancouver BC V6T 1Z2
Canada
tbjw@math.ubc.ca
Abstract.

We consider the general circumstance of an Azumaya algebra A𝐴A of degree n𝑛n over a locally ringed topos (X,𝒪X)Xsubscript𝒪X(\text{\bf X},\mathcal{O}_{\text{\bf X}}) where the latter carries a (possibly trivial) involution, denoted λ𝜆\lambda. This generalizes the usual notion of involutions of Azumaya algebras over schemes with involution, which in turn generalizes the notion of involutions of central simple algebras. We provide a criterion to determine whether two Azumaya algebras with involutions extending λ𝜆\lambda are locally isomorphic, describe the equivalence classes obtained by this relation, and settle the question of when an Azumaya algebra A𝐴A is Brauer equivalent to an algebra carrying an involution extending λ𝜆\lambda, by giving a cohomological condition. We remark that these results are novel even in the case of schemes, since we allow ramified, non-trivial involutions of the base object. We observe that, if the cohomological condition is satisfied, then A𝐴A is Brauer equivalent to an Azumaya algebra of degree 2n2𝑛2n carrying an involution. By comparison with the case of topological spaces, we show that the integer 2n2𝑛2n is minimal, even in the case of a nonsingular affine variety X𝑋X with a fixed-point free involution. As an incidental step, we show that if R𝑅R is a commutative ring with involution for which the fixed ring S𝑆S is local, then either R𝑅R is local or R/S𝑅𝑆R/S is a quadratic étale extension of rings.

2010 Mathematics Subject Classification:
Primary: 16H05, 14F22, 11E39, 55P91
First was supported, in part, by a UBC postdoctoral fellowship.

1. Introduction

1.1. Motivation

Let A𝐴A be a central simple algebra over a field K𝐾K and let τ:AA:𝜏𝐴𝐴\tau:A\to A be an involution, i.e., an anti-automorphism satisfying aττ=asuperscript𝑎𝜏𝜏𝑎a^{\tau\tau}=a for all aA𝑎𝐴a\in A. Recall that τ𝜏\tau can be of the first kind or of the second kind, depending on whether τ𝜏\tau restricts to the identity on the centre K𝐾K or not. We further say that τ𝜏\tau is a λ𝜆\lambda-involution where λ=τ|K𝜆evaluated-at𝜏𝐾\lambda=\tau|_{K}.

Central simple algebras and their involutions play a major role in the theory of classical algebraic groups, and also in Galois cohomology. For example, letting F𝐹F denote the fixed field of λ:KK:𝜆𝐾𝐾\lambda:K\to K, it is well-known that the absolutely simple adjoint classical algebraic groups over F𝐹F are all given as the neutral connected component of projective unitary groups of algebras with involution (A,τ)𝐴𝜏(A,\tau) as above, where K𝐾K varies (here we also allow K=F×F𝐾𝐹𝐹K=F\times F with the switch involution), see [knus_book_1998-1, §26]. In fact, all simple algebraic groups of types A𝐴A, B𝐵B, C𝐶C, D𝐷D, excluding D4subscript𝐷4D_{4}, can be described by means of central simple algebras with involution. Involutions of central simple algebras also arise naturally in representation theory, either since group algebras admit a canonical involution, or in the context of orthogonal, unitary, or symplectic representations, see, for instance, [riehm_orthogonal_rep_2001].

Azumaya algebras are generalizations of central simple algebras in which the base field is replaced with a ring, or more generally, a scheme. As with central simple algebras, Azumaya algebras and their involutions are important in the study of classical reductive group schemes, as well as in étale cohomology and in the representation theory of finite groups over rings; see [knus_quadratic_1991].

Suppose that K/F𝐾𝐹K/F is a quadratic Galois extension of fields and let λ𝜆\lambda denote the non-trivial F𝐹F-automorphism of K𝐾K. A theorem of Albert, Riehm and Scharlau, [knus_book_1998-1, Thm. 3.1(2)], asserts that a central simple K𝐾K-algebra A𝐴A admits a λ𝜆\lambda-involution if and only if [A]delimited-[]𝐴[A], the Brauer class of A𝐴A, lies in the kernel of the corestriction map coresK/F:Br(K)Br(F):subscriptcores𝐾𝐹Br𝐾Br𝐹\operatorname{cores}_{K/F}:\operatorname{Br}(K)\to\operatorname{Br}(F). Saltman [saltman_azumaya_1978, Thm. 3.1b] later showed that if K/F𝐾𝐹K/F is replaced with a quadratic Galois extension of rings R/S𝑅𝑆R/S, then the class [A]delimited-[]𝐴[A] lies in the kernel of coresR/S:Br(R)Br(S):subscriptcores𝑅𝑆Br𝑅Br𝑆\operatorname{cores}_{R/S}:\operatorname{Br}(R)\to\operatorname{Br}(S) if and only if some representative A[A]superscript𝐴delimited-[]𝐴A^{\prime}\in[A] admits a λ𝜆\lambda-involution. Here, in distinction to the case of fields, an arbitrary representative may not posses an involution. However, a later proof by Knus, Parimala and Srinivas [knus_azumaya_1990, Th. 4.2], which applies to Azumaya algebras over schemes, implies that one can take A[A]superscript𝐴delimited-[]𝐴A^{\prime}\in[A] such that degA=2degAdegsuperscript𝐴2deg𝐴\operatorname{deg}A^{\prime}=2\operatorname{deg}A.

The aforementioned results all have counterparts for involutions of the first kind in which the condition coresR/S[A]=0subscriptcores𝑅𝑆𝐴0\operatorname{cores}_{R/S}[A]=0 is replaced by 2[A]=02delimited-[]𝐴02[A]=0.

In this article, we generalize this theory to more general sites and more general involutions. We have two purposes in doing so. The first, our initial motivation, is to demonstrate that the upper bound in Saltman’s theorem, degA2degAdegsuperscript𝐴2deg𝐴\operatorname{deg}A^{\prime}\leq 2\operatorname{deg}A guaranteed by [knus_azumaya_1990, Th. 4.2], cannot be improved in general for involutions of the second kind. The statement in the case of involutions of the first kind was established in [asher_auel_azumaya_2017]. Our general approach here is similar to [asher_auel_azumaya_2017], [antieau_unramified_2014] and related works. That is, the desired example is constructed by approximating a suitable classifying space, and topological obstruction theory is used to show that it has the required properties. In contrast with [asher_auel_azumaya_2017] and [antieau_unramified_2014], the obstruction is obtained by means of equivariant homotopy theory.

We therefore introduce and study involutions of the second kind of Azumaya algebras on topological spaces. In fact, we develop the necessary foundations in the generality of connected locally ringed topoi with involution, and show that Saltman’s theorem holds in this setting. In doing so, we stumbled into our second purpose, which we now explain.

Any involution of a field is either trivial or comes from a quadratic Galois extension, which is why the classical theory sees a dichotomy into involutions of the first or second kind. For a ring, the analogous involutions are the trivial involutions or those arising as the non-trivial automorphism of a quadratic étale extension R/S𝑅𝑆R/S. Geometrically, these correspond to extreme cases where one has either a trivial action of the cyclic group C2={1,λ}subscript𝐶21𝜆C_{2}=\{1,\lambda\} on a scheme, or where the action is scheme-theoretically free. One may also view the free case as corresponding to an unramified map π:XX/C2:𝜋𝑋𝑋subscript𝐶2\pi:X\to X/C_{2}. This dichotomy has been preserved in the literature on involutions of Azumaya algebras over schemes, say for instance [knus_azumaya_1990] and [knus_quadratic_1991], by considering only trivial or unramified involutions of the base ring.

There are, of course, involutions λ:RR:𝜆𝑅𝑅\lambda:R\to R which are neither trivial nor wholly unramified. For instance, one may encounter involutions of varieties that are generically free but fix a nonempty closed subscheme. Alternatively, there are involutions of nonreduced rings that restrict to trivial involutions of the reduction—these are geometrically ramified everywhere, but nonetheless non-trivial.

Our second purpose therefore became developing the theory of λ𝜆\lambda-involutions on Azumaya algebras with minimal assumptions on λ𝜆\lambda. We establish a generalization of Saltman’s theorem, and present a classification of λ𝜆\lambda-involutions into types, generalizing the classification of involutions of central simple algebras as orthogonal, symplectic—both of the first kind—or unitary—of the second.

In more detail, given a field K𝐾K of characteristic not 222 and an involution λ:KK:𝜆𝐾𝐾\lambda:K\to K, recall that two degree-n𝑛n central simple K𝐾K-algebras with involutions extending λ𝜆\lambda are of the same type if they become isomorphic after base change to a separable closure of the fixed field of λ𝜆\lambda. This definition extends naturally to the case of a general connected ring R𝑅R in which 222 is a unit by replacing “a separable closure” by an étale extension of S𝑆S, the fixed ring of λ:RR:𝜆𝑅𝑅\lambda:R\to R. It is natural to ask how many types are obtained in this manner, and how to distinguish them effectively. In the classically-considered cases of trivial or unramified involutions on R𝑅R, the situation is known to be similar to case of fields: When R=S𝑅𝑆R=S, there are at most two types — the orthogonal, which occurs for all n𝑛n, and the symplectic, which occurs only for even n𝑛n. When R/S𝑅𝑆R/S is quadratic étale, only one type, called the unitary type, occurs for all n𝑛n.

We describe the types for arbitrary λ:RR:𝜆𝑅𝑅\lambda:R\to R and give a cohomological criterion to determine when two involutions are of the same type. This criterion implies in particular that the type of an Azumaya algebra with involution (A,τ)𝐴𝜏(A,\tau) is determined entirely by the restriction of (A,τ)𝐴𝜏(A,\tau) to the ramification locus of SpecRSpecSSpec𝑅Spec𝑆\operatorname{Spec}R\to\operatorname{Spec}S. More than two types may occur. With this new subtlety, one can further ask, in the context of Saltman’s theorem, what are the types of λ𝜆\lambda-involutions which can be exhibited on representatives of a given Brauer class in BrRBr𝑅\operatorname{Br}R. Our generalization of Saltman’s theorem answers this question.

To demonstrate some of the ideas above, let us consider a field k𝑘k of characteristic different from 222 and the ring R:=k[x,x1]assign𝑅𝑘𝑥superscript𝑥1R:=k[x,x^{-1}] of Laurent polynomials with the involution λ:xx1:𝜆maps-to𝑥superscript𝑥1\lambda:x\mapsto x^{-1}. The fixed ring is S:=k[x+x1]assign𝑆𝑘delimited-[]𝑥superscript𝑥1S:=k[x+x^{-1}]. The map SpecRSpecSSpec𝑅Spec𝑆\operatorname{Spec}R\to\operatorname{Spec}S is ramified at two points, x=1𝑥1x=1 and x=1𝑥1x=-1, and unramified elsewhere. Our results show that there are 444 types of λ𝜆\lambda-involution for even-degree algebras and 111 type in odd degrees. Furthermore, the type of a λ𝜆\lambda-involution is determined by the types — orthogonal or symplectic — obtained by specializing to x=1𝑥1x=1 and x=1𝑥1x=-1. For example, consider the λ𝜆\lambda-involution of Mat2×2(R)subscriptMat22𝑅\operatorname{Mat}_{2\times 2}(R) given by

(1) τ:[a(x)b(x)c(x)d(x)][d(x1)x1b(x1)xc(x1)a(x1)].:𝜏maps-tomatrix𝑎𝑥𝑏𝑥𝑐𝑥𝑑𝑥matrix𝑑superscript𝑥1superscript𝑥1𝑏superscript𝑥1𝑥𝑐superscript𝑥1𝑎superscript𝑥1\tau:\begin{bmatrix}a(x)&b(x)\\ c(x)&d(x)\end{bmatrix}\mapsto\begin{bmatrix}d(x^{-1})&x^{-1}b(x^{-1})\\ xc(x^{-1})&a(x^{-1})\end{bmatrix}.

Evaluating at x=1𝑥1x=1, the involution of (1) becomes orthogonal, whereas evaluating at x=1𝑥1x=-1 makes it symplectic. Our generalization of Saltman’s theorem implies that if αBrR𝛼Br𝑅\alpha\in\operatorname{Br}R is represented by an Azumaya R𝑅R-algebra admitting a λ𝜆\lambda-involution, then each of the 444 types of λ𝜆\lambda-involutions is the type of a λ𝜆\lambda-involution of some representative of α𝛼\alpha.

It seems likely that our results on types could be used to extend the theory of involutive Brauer groups, intiated in [parimala_92] (see also [verschoren_98]), to schemes carrying ramified involutions. We hope to address this in subsequent work.

We finally note that from the point of view of group schemes, the study of λ𝜆\lambda-involutions of Azumaya algebras in the case where is λ𝜆\lambda neither trivial nor unramified amounts to studying certain group schemes over SpecRSpec𝑅\operatorname{Spec}R which are generically reductive but degenerate on a divisor. Specifically, the projective unitary group of an Azumaya algebra with a λ𝜆\lambda-involution is generically of type A𝐴A and degenerates to types B𝐵B, C𝐶C or D𝐷D on the connected components of the branch locus of SpecRSpecSSpec𝑅Spec𝑆\operatorname{Spec}R\to\operatorname{Spec}S. The study of degenerations of reductive groups have proved useful in many instances. Recent examples include [auel_parimala_suresh_2015] and [bayer_17], but this manifests even more in the works of Bruhat and Tits on reductive groups over henselian discretely valued fields [bruhat_I_72], [bruhat_II_84], [bruhat_III_87]. Broadly speaking, degenerations of reductive groups are encountered naturally when one attempts to extend a group scheme defined on a generic point of an integral scheme to the entire scheme, a process which is often considered in number theory.

1.2. Outline

Following is a detailed account of the contents of this paper, mostly in the order of presentation. While the majority of this work applies to schemes without assuming 222 is invertible, we make this assumption here in order to avoid certain technicalities.

Section 2 is devoted to technical preliminaries, largely to do with non-abelian cohomology in the context of Gorthendieck topoi.

Let X𝑋X be a scheme and let λ:XX:𝜆𝑋𝑋\lambda:X\to X be an involution. Our first concern is to specify an appropriate quotient of X𝑋X by the group C2={1,λ}subscript𝐶21𝜆C_{2}=\{1,\lambda\}. There is an evident choice when X=SpecR𝑋Spec𝑅X=\operatorname{Spec}R with R𝑅R a ring, since one can take the quotient to be SpecSSpec𝑆\operatorname{Spec}S, where S𝑆S is the fixed ring of λ:RR:𝜆𝑅𝑅\lambda:R\to R. However, at the level of generality that we consider, there is often more than one plausible option. For instance, if the action of λ𝜆\lambda is not free, then [X/C2]delimited-[]𝑋subscript𝐶2[X/C_{2}], a Deligne–Mumford stack, might serve just as well as the scheme or algebraic space X/C2𝑋subscript𝐶2X/C_{2}. The difference between these alternatives becomes particularly striking when C2subscript𝐶2C_{2} acts trivially on X𝑋X — the quotient XX/C2=X𝑋𝑋subscript𝐶2𝑋X\to X/C_{2}=X can be regarded as a degenerate case of a double covering, ramified everywhere, whereas X[X/C2]𝑋delimited-[]𝑋subscript𝐶2X\to[X/C_{2}] is a C2subscript𝐶2C_{2}-Galois covering, ramified nowhere. From the point of view of the first quotient, all involutions will appear to be of the first kind, whereas with respect to the second quotient, all involutions will appear to be of the second kind.

We are therefore led to conclude that a chosen quotient π:XY:𝜋𝑋𝑌\pi:X\to Y, in addition to X𝑋X and λ𝜆\lambda, is necessary in order to discuss involutions in a way consistent with what is already done in the cases where λ𝜆\lambda is an involution of a ring.

We require a quotient to satisfy certain axioms, presented in Subsection 4.3, and prove that they are satisfied in a number of important examples, notably when the categorical quotient X/C2𝑋subscript𝐶2X/C_{2} exists in the category of schemes and is a good quotient. Such quotients exist for instance if X𝑋X is affine or projective, see Theorem 4.4.4. Thereafter in the development of the theory, we are usually agnostic about the quotient chosen. In examples, we often return to the motivating case of a good quotient.

Consider, therefore, a good quotient π:XY=X/C2:𝜋𝑋𝑌𝑋subscript𝐶2\pi:X\to Y=X/C_{2}. It is technically easier to work on Y𝑌Y than on X𝑋X. Specifically, by virtue of our Theorem 4.3.11, there is an equivalence between Azumaya algebras with λ𝜆\lambda-involution on X𝑋X on the one hand and Azumaya algebras with πλsubscript𝜋𝜆\pi_{*}\lambda-involution over the sheaf of rings R:=π(𝒪X)assign𝑅subscript𝜋subscript𝒪𝑋R:=\pi_{*}(\mathcal{O}_{X}) on the other. We therefore study Azumaya algebras over R𝑅R. While Y𝑌Y does not carry an involution, the ring sheaf R𝑅R has an involution, namely, πλsubscript𝜋𝜆\pi_{*}\lambda, which we abbreviate to λ𝜆\lambda. A difficulty that we encounter here is that the sheaf of rings R𝑅R is not a local ring object on Y𝑌Y, but rather a sheaf of rings with involution, the fixed subsheaf of which is the local ring object (π𝒪X)C2=𝒪Ysuperscriptsubscript𝜋subscript𝒪𝑋subscript𝐶2subscript𝒪𝑌(\pi_{*}\mathcal{O}_{X})^{C_{2}}=\mathcal{O}_{Y}. We devote considerable work to the study of commutative rings with involutions whose fixed subrings are local in Section 3, and conclude in Theorem 3.3.8 that any such ring is a semilocal ring, so that the sheaf R𝑅R may be viewed as making Y𝑌Y a “semilocally ringed” space.

In Section 5, we introduce and study types of λ𝜆\lambda-involutions. Specifically, we define two Azumaya R𝑅R-algebras with a λ𝜆\lambda-involution, (A,τ)𝐴𝜏(A,\tau) and (B,σ)𝐵𝜎(B,\sigma), to be of the same type if some matrix algebra over (A,τ)𝐴𝜏(A,\tau) is Yétsubscript𝑌étY_{\text{\'{e}t}}-locally isomorphic to some matrix algebra over (B,σ)𝐵𝜎(B,\sigma). We show in Theorem 5.2.13 and Corollary 5.2.14 that the collection of types forms a 222-torsion group whose product rule is compatible with tensor products, and when degA=degBdeg𝐴deg𝐵\operatorname{deg}A=\operatorname{deg}B, the involutions τ𝜏\tau and σ𝜎\sigma have the same type if and only if (A,τ)𝐴𝜏(A,\tau) and (B,σ)𝐵𝜎(B,\sigma) are Yétsubscript𝑌étY_{\text{\'{e}t}}-locally isomorphic, without the need to pass to matrix algebras. Thus, the definition given here agrees with the definition in Subsection 1.1. We then turn to the problem of calculating the group of types in specific cases.

Let WY𝑊𝑌W\subset Y denote the branch locus of π:XY:𝜋𝑋𝑌\pi:X\to Y. Then, away from W𝑊W, the C2subscript𝐶2C_{2}-action on V=Xπ1(W)𝑉𝑋superscript𝜋1𝑊V=X-\pi^{-1}(W) is unramified, hence there is only one possible type of λ𝜆\lambda-involution on A|Vevaluated-at𝐴𝑉A|_{V}, viz. unitary, and all involutions on A|Vevaluated-at𝐴𝑉A|_{V} are locally isomorphic to the involution Matn×n(R)Matn×n(R)subscriptMat𝑛𝑛𝑅subscriptMat𝑛𝑛𝑅\operatorname{Mat}_{n\times n}(R)\to\operatorname{Mat}_{n\times n}(R) given by applying the involution λ𝜆\lambda to each entry in the matrix and then taking the transpose, i.e., M(Mλ)trmaps-to𝑀superscriptsuperscript𝑀𝜆trM\mapsto(M^{\lambda})^{\text{\rm tr}}. In contrast, over a connected component Z1subscript𝑍1Z_{1} of Z:=π1(W)assign𝑍superscript𝜋1𝑊Z:=\pi^{-1}(W), regarded as a reduced closed subscheme of X𝑋X, the involution λ𝜆\lambda restricts to the identity (Proposition 4.5.5), and so λ𝜆\lambda-involutions of A|Z1evaluated-at𝐴subscript𝑍1A|_{Z_{1}} fall into one of two types — orthogonal or symplectic. This suggests that the types of λ𝜆\lambda-involutions over X𝑋X should be in bijection with H0(Z,μ2)superscriptH0𝑍subscript𝜇2\mathrm{H}^{0}(Z,\mu_{2}), where μ2:={1,1}assignsubscript𝜇211\mu_{2}:=\{1,-1\} and 111 and 11-1 represent orthogonal and symplectic involutions respectively, and that two λ𝜆\lambda-involutions are of the same type if and only if they are of the same type when restricted to each connected component of Z𝑍Z. We prove the second statement in Theorem 5.4.5 and establish the first under the assumption that that Y𝑌Y is noetherian and regular in Corollary 6.4.8. We do not know whether the first statement holds in general. Determining the type of a given involution of a given algebra, τ:AA:𝜏𝐴𝐴\tau:A\to A, can now be carried out by considering the rank of the sheaf of τ𝜏\tau-symmetric elements on the various components of W𝑊W; see [knus_book_1998-1, Prp. 2.6].

In Section 6, we turn to the question of when a Brauer class α=[A]Br(X)𝛼delimited-[]𝐴Br𝑋\alpha=[A]\in\operatorname{Br}(X) contains an algebra Asuperscript𝐴A^{\prime} possessing a λ𝜆\lambda-involution. Saltman [saltman_azumaya_1978, Thm. 3.1] gave necessary and sufficient conditions for this when λ𝜆\lambda is trivial or unramified. Specifically, A𝐴A is equivalent to such an algebra if 2[A]=0Br(X)2delimited-[]𝐴0Br𝑋2[A]=0\in\operatorname{Br}(X), in the case of a trivial action, or if coresX/Y[A]=0Br(Y)subscriptcores𝑋𝑌𝐴0Br𝑌\operatorname{cores}_{X/Y}[A]=0\in\operatorname{Br}(Y), in the case of an unramified action. We unify these two results, and generalize to the cases that are neither trivial nor unramified, by defining a transfer map transf:Br(X)Hét2(Y,𝔾m):transfBr𝑋superscriptsubscriptHét2𝑌subscript𝔾𝑚\operatorname{transf}:\operatorname{Br}(X)\to\mathrm{H}_{\text{\'{e}t}}^{2}(Y,\mathbb{G}_{m}), and deducing in Theorem 6.3.3 that A𝐴A is equivalent to an algebra admitting a λ𝜆\lambda-involution of type t𝑡t if and only if transf([A])=Φ(t)transfdelimited-[]𝐴Φ𝑡\operatorname{transf}([A])=\Phi(t), where Φ(t)H2(Y,𝔾m)Φ𝑡superscriptH2𝑌subscript𝔾𝑚\Phi(t)\in\mathrm{H}^{2}(Y,\mathbb{G}_{m}) is a cohomology class depending on the type. In both extreme cases of trivial and unramified actions, and in fact whenever Y𝑌Y is a nonsingular variety, Φ(t)Φ𝑡\Phi(t) is necessarily 00. Moreover, in the case of a trivial action, transf([A])=2[A]transfdelimited-[]𝐴2delimited-[]𝐴\operatorname{transf}([A])=2[A], and in the unramified case, transf([A])=coresX/Y[A]transfdelimited-[]𝐴subscriptcores𝑋𝑌𝐴\operatorname{transf}([A])=\operatorname{cores}_{X/Y}[A], so we recover Saltman’s theorem as a special case. We also show that if A𝐴A is equivalent to an algebra with involution, then such an algebra can be constructed to have degree twice that of A𝐴A, thus extending the analogous result of [knus_azumaya_1990, Thms. 4.1, 4.2]. We do not, however, follow [saltman_azumaya_1978] and [knus_azumaya_1990] in considering the corestriction algebra of A𝐴A, taking instead a purely cohomological approach. In fact, it is not clear whether a corestriction algebra of A𝐴A can be defined in a meaningful way when λ:XX:𝜆𝑋𝑋\lambda:X\to X is ramified. This problem was considered in [auel_parimala_suresh_2015, §5], where some positive results are given, and we leave its pursuit in the current level of generality to a future work.

Section 7 gives a number of examples of the workings out of the previous theory. In particular, we give examples of schemes X𝑋X with involutions λ:XX:𝜆𝑋𝑋\lambda:X\to X that are neither unramified nor trivial, along with a classification of the various types of λ𝜆\lambda-involutions of Azumaya algebras, e.g., Examples 7.1.6 and 7.1.7.

While this overview has so far been written in the language of schemes, the majority of the results are established in the setting of locally ringed Grothendieck topoi, of which the étale ringed topoi of a scheme is a special case. The advantage of this generality is that all the results above also apply, essentially verbatim, to Azumaya algebras with involution over a topological C2subscript𝐶2C_{2}-space, or to Azumaya algebras with involutions on algebraic stacks. The applicability of our results in the context of other sites associated with schemes, e.g., the Zariski site, the fppf site, the Nisnevich site and some large sites, is discussed in Subsection 4.4.

Comparison of Azumaya algebras over schemes with topological Azumaya algebras has proved useful in the past, for instance in [antieau_unramified_2014], [antieau_topology_2015]. Having the previous theory available also in the topological context, we consider a finite type, regular \mathbb{C}-algebra R𝑅R with an unramified involution λ𝜆\lambda and compare the theory of Azumaya R𝑅R-algebras with involutions restricting to λ𝜆\lambda on the centre with the theory of topological Azumaya algebras with involution on the complex manifold (SpecR)ansubscriptSpec𝑅an(\operatorname{Spec}R)_{\text{an}}. This is carried out in Subsection 4.2, specifically in Example 4.2.4.

By such comparison, we produce an example of an Azumaya algebra A𝐴A of degree n𝑛n, over a ring R𝑅R with an unramified involution λ𝜆\lambda, having the property that A𝐴A is Brauer equivalent to an algebra Asuperscript𝐴A^{\prime} with λ𝜆\lambda-involution, but the least degree of such an Asuperscript𝐴A^{\prime} is 2n2𝑛2n, Theorem 9.2.3; the bound 2n2𝑛2n is the lowest possible by [knus_azumaya_1990, §4], which guarantees the existence of Asuperscript𝐴A^{\prime} of degree 2n2𝑛2n in general. An analogous example in the case where λ𝜆\lambda is assumed to be trivial was given in [asher_auel_azumaya_2017]. The method of proof, which is carried out in Sections 8 and 9, is by using existing study of bundles with involution as a branch of equivariant homotopy theory, [may_equivariant_1996]. In particular, we can find universal examples of topological Azumaya algebras with involution, which are valuable sources of counterexamples.

In an appendix, we give a proof that the stalks of the sheaf of continuous, complex-valued functions on a topological space X𝑋X satisfy Hensel’s lemma. This is used here and there in the body of the paper to treat this case at the same time as étale sites of schemes.

1.3. Acknowledgments

The authors would like to thank Zinovy Reichstein for introducing them to each other and recommending that they study involutions of Azumaya algebras from a topological point of view. They would like to thank Asher Auel for helpful conversations and good ideas, some of which appear in this paper. They owe an early form of an argument in 5.4 to Sune Precht Reeh. The second author would like to thank Omar Antolín, Akhil Mathew, Mona Merling, Marc Stephan and Ric Wade for various conversations about equivariant classifying spaces, and Bert Guillou for a reference to the literature on equivariant model structures. The second author would like to thank Ben Antieau for innumerable valuable conversations about Azumaya algebras from the topological point of view, and would like to thank Gwendolyn Billett for help in deciphering [giraud_cohomologie_1971]. We also thank the referees for many valuable suggestions.

2. Preliminaries

This section recalls necessary facts and sets notation for the sequel. Throughout, 𝐗𝐗{\mathbf{X}} denotes a Grothendieck topos. We reserve the term “ring” for commutative unital rings, whereas algebras are assumed unital but not necessarily commutative.

2.1. Generalities on Topoi

Recall that a Grothendieck topos is a category that is equivalent to the category of set-valued sheaves over a small site, or equivalently, a category satisfying Giraud’s axioms; see [giraud_cohomologie_1971, Chap. 0]. In this paper we shall be particularly interested in the following examples:

  1. (i)

    𝐗=Sh(Xét)𝐗Shsubscript𝑋ét{\mathbf{X}}=\text{\bf Sh}(X_{\text{\'{e}t}}), the category of sheaves over the small étale site of a scheme X𝑋X.

  2. (ii)

    𝐗=Sh(X)𝐗Sh𝑋{\mathbf{X}}=\text{\bf Sh}(X), the category of sheaves on a topological space X𝑋X.

We will occasionally consider other sites associated with a scheme X𝑋X. In particular, XZarsubscript𝑋ZarX_{\operatorname{Zar}} and Xfppfsubscript𝑋fppfX_{\mathrm{fppf}} will denote the small Zariski and small fppf sites of X𝑋X, respectively.

The topos of sheaves over a singleton topological space, which is nothing but the category of sets, will be denoted 𝐩𝐭𝐩𝐭\mathbf{pt}.

We note that every topos 𝐗𝐗\mathbf{X} can be regarded as a site relative to its canonical topology. In this case, a collection of morphisms {UiV}iIsubscriptsubscript𝑈𝑖𝑉𝑖𝐼\{U_{i}\to V\}_{i\in I} is a covering of V𝑉V if and only if it is jointly surjective, and every sheaf over 𝐗𝐗\mathbf{X} is representable, so that 𝐗Sh(𝐗)𝐗Sh𝐗\mathbf{X}\cong\text{\bf Sh}(\mathbf{X}). This allows us to define objects of 𝐗𝐗{\mathbf{X}} by specifying the sheaf that they represent, and to define morphisms between objects by defining them on sections.

The symbols 𝐗subscript𝐗\emptyset_{{\mathbf{X}}} and 𝐗subscript𝐗*_{\mathbf{X}} will be used for the initial and final objects of 𝐗𝐗\mathbf{X}, respectively. When 𝐗=Sh(X)𝐗Sh𝑋\mathbf{X}=\text{\bf Sh}(X) for a site X𝑋X, the sheaf 𝐗subscript𝐗\emptyset_{{\mathbf{X}}} assigns an empty set to every non-initial object of X𝑋X, and 𝐗subscript𝐗*_{{\mathbf{X}}} is the sheaf assigning a singleton to every object in X𝑋X. The subscript 𝐗𝐗{\mathbf{X}} will dropped when it may be understood from the context.

For every object A,U𝐴𝑈A,U of 𝐗𝐗{\mathbf{X}} the U𝑈U-sections of A𝐴A are

A(U):=Hom𝐗(U,A)assign𝐴𝑈subscriptHom𝐗𝑈𝐴A(U):=\operatorname{Hom}_{\mathbf{X}}(U,A)

and the global sections of A𝐴A are H0(𝐗,A)=ΓA=Γ𝐗A:=A()superscriptH0𝐗𝐴Γ𝐴subscriptΓ𝐗𝐴assign𝐴\mathrm{H}^{0}(\mathbf{X},A)=\Gamma A=\Gamma_{\mathbf{X}}A:=A(*). We will write AU=A×Usubscript𝐴𝑈𝐴𝑈A_{U}=A\times U, and will regard AUsubscript𝐴𝑈A_{U} as an object of the slice category 𝐗/U𝐗𝑈\mathbf{X}/U.

By a group G𝐺G in 𝐗𝐗\mathbf{X} we will mean a group object in 𝐗𝐗\mathbf{X}. In this case, the U𝑈U-sections G(U)𝐺𝑈G(U) form a group for all objects U𝑈U of 𝐗𝐗\mathbf{X}. Similar conventions will apply to abelian groups, rings, G𝐺G-objects, and so on.

If R𝑅R is a ring object in some topos, then μ2,Rsubscript𝜇2𝑅\mu_{2,R} will denote the object of square-roots of 111 in R𝑅R, that is, the object given section-wise by μ2,R(U)={xR(U):x2=1}subscript𝜇2𝑅𝑈conditional-set𝑥𝑅𝑈superscript𝑥21\mu_{2,R}(U)=\{x\in R(U)\,:\,x^{2}=1\}. The bald notation μ2subscript𝜇2\mu_{2} will denote the constant sheaf {+1,1}11\{+1,-1\}.

2.2. Torsors

Definition 2.2.1.

Let X𝑋X be a site and let G𝐺G be a sheaf of groups on X𝑋X. A (right) G𝐺G-torsor is a sheaf P𝑃P on X𝑋X equipped with a right action P×GP𝑃𝐺𝑃P\times G\to P such that P𝑃P is locally isomorphic to G𝐺G as a right G𝐺G-object.

Equivalently, and intrinsically to the topos 𝐗:=Sh(X)assign𝐗Sh𝑋\mathbf{X}:=\text{\bf Sh}(X), a (right) G𝐺G-torsor is an object P𝑃P of 𝐗𝐗\mathbf{X} equipped with a (right) G𝐺G-action m:P×GP:𝑚𝑃𝐺𝑃m:P\times G\to P such that the unique morphism P𝑃P\to\ast is an epimorphism and such that the morphism P×Gπ1×mP×P𝑃𝐺subscript𝜋1𝑚𝑃𝑃P\times G\overset{\pi_{1}\times m}{\longrightarrow}P\times P is an isomorphism. See [giraud_cohomologie_1971, Déf. III.1.4.1] where more general torsors over objects S𝑆S of 𝐗𝐗\mathbf{X} are defined; our definition is that of torsors over the terminal object.

The equivalence of the two definitions of “torsor” is given by [giraud_cohomologie_1971, Prop. III.1.7.3].

The category of G𝐺G-torsors, with G𝐺G-equivariant isomorphisms as morphisms, will be denoted

Tors(𝐗,G).Tors𝐗𝐺\text{\bf Tors}({\mathbf{X}},G).

A G𝐺G-torsor P𝑃P is trivial if PG𝑃𝐺P\cong G as right G𝐺G-objects, and an object U𝑈U is said to trivialize P𝑃P if PUGUsubscript𝑃𝑈subscript𝐺𝑈P_{U}\cong G_{U} as GUsubscript𝐺𝑈G_{U}-objects. The latter holds precisely when P(U)𝑃𝑈P(U)\neq\emptyset.

Recall that if P𝑃P is a G𝐺G-torsor and X𝑋X is a left G𝐺G-object in 𝐗𝐗{\mathbf{X}}, then P×GXsuperscript𝐺𝑃𝑋P\times^{G}X denotes the quotient of P×X𝑃𝑋P\times X by the equivalence relation P×G×X(P×X)×(P×X)𝑃𝐺𝑋𝑃𝑋𝑃𝑋P\times G\times X\to(P\times X)\times(P\times X) given by (p,g,x)((pg,x),(p,gx))maps-to𝑝𝑔𝑥𝑝𝑔𝑥𝑝𝑔𝑥(p,g,x)\mapsto((pg,x),(p,gx)) on sections. We shall sometimes denote P×GXsuperscript𝐺𝑃𝑋P\times^{G}X by XPsuperscript𝑋𝑃{}^{P}X and call it the P𝑃P-twist of X𝑋X. We remark that X𝑋X and XPsuperscript𝑋𝑃{}^{P}X are locally isomorphic in the sense that there exists a covering U𝑈U\to* in 𝐗𝐗{\mathbf{X}} such that XUXUPsubscript𝑋𝑈superscriptsubscript𝑋𝑈𝑃X_{U}\cong{}^{P}X_{U} — take any U𝑈U such that GUPUsubscript𝐺𝑈subscript𝑃𝑈G_{U}\cong P_{U}. If X𝑋X posses some additional structure, for instance if X𝑋X is an abelian group, and G𝐺G respects this structure, then XPsuperscript𝑋𝑃{}^{P}X also posses the same structure and the isomorphism XUXUPsubscript𝑋𝑈superscriptsubscript𝑋𝑈𝑃X_{U}\cong{}^{P}X_{U} respects the additional structure. The general theory outlined here is established precisely in [giraud_cohomologie_1971, Chap. III].

Remark 2.2.2.

There is another plausible definition of “torsor” on a site X𝑋X, particularly when the topology is subcanonical and when the category X𝑋X has finite products—i.e., X𝑋X is a standard site. That is, one modifies the definition in 2.2.1 by requiring the objects G𝐺G and P𝑃P to be objects of the site X𝑋X. These are the representable torsors as distinct from the sheaf torsors defined above. We will not consider the question of representability in this paper beyond the following remark: Suppose X𝑋X is a scheme and G𝐺G is a group scheme over X𝑋X. Then G𝐺G represents a group sheaf on the big flat site of X𝑋X, also denoted G𝐺G. If GX𝐺𝑋G\to X is affine, then all sheaf G𝐺G-torsors are representable by an X𝑋X-scheme [milne_etale_1980, Thm. III.4.3].

2.3. Cohomology of Abelian Groups

The functor H0superscriptH0\mathrm{H}^{0} sending an abelian group A𝐴A in 𝐗𝐗\mathbf{X} to its global sections is left exact. The i𝑖i-th right derived functor of H0superscriptH0\mathrm{H}^{0} is denoted Hi(𝐗,A)superscriptH𝑖𝐗𝐴\mathrm{H}^{i}(\mathbf{X},A), as usual. If 𝐗𝐗\mathbf{X} is clear from the context, we shall simply write Hi(A)superscriptH𝑖𝐴\mathrm{H}^{i}(A). When 𝐗=Sh(Xét)𝐗Shsubscript𝑋ét\mathbf{X}=\text{\bf Sh}(X_{\text{\'{e}t}}) for a scheme X𝑋X, we write Hi(𝐗,A)superscriptH𝑖𝐗𝐴\mathrm{H}^{i}(\mathbf{X},A) as Héti(X,A)subscriptsuperscriptH𝑖ét𝑋𝐴\mathrm{H}^{i}_{\text{\'{e}t}}(X,A), and likewise for other sites associated with X𝑋X.

In the sequel, we shall make repeated use of Verdier’s Theorem, quoted below, which provides a description of cohomology classes in terms of hypercoverings. We recall some details, and in doing so, we set notation. One may additionally consult [de_jong_stacks_2017, Tag 01FX], [dugger_hypercovers_2004] or [artin_theorie_1972, Exp. V.7].

Let 𝚫𝚫{\bm{\Delta}} denote the category having {{0,,n}|n=0,1,2,}conditional-set0𝑛𝑛012\{\{0,\dots,n\}\,|\,n=0,1,2,\dots\} as its objects and the non-decreasing functions as its morphisms. Recall that a simplicial object in 𝐗𝐗{\mathbf{X}} is a contravariant functor U:𝚫𝐗:subscript𝑈𝚫𝐗U_{\bullet}:{\bm{\Delta}}\to{\mathbf{X}}. For every 0in0𝑖𝑛0\leq i\leq n, we write Un=U({0,,n})subscript𝑈𝑛subscript𝑈0𝑛U_{n}=U_{\bullet}(\{0,\dots,n\}) and set din=U(δin)subscriptsuperscript𝑑𝑛𝑖subscript𝑈subscriptsuperscript𝛿𝑛𝑖d^{n}_{i}=U_{\bullet}(\delta^{n}_{i}) and sin=U(σin)subscriptsuperscript𝑠𝑛𝑖subscript𝑈superscriptsubscript𝜎𝑖𝑛s^{n}_{i}=U_{\bullet}(\sigma_{i}^{n}), where δin:{0,,n1}{0,,n}:subscriptsuperscript𝛿𝑛𝑖0𝑛10𝑛\delta^{n}_{i}:\{0,\dots,n-1\}\to\{0,\dots,n\} is the non-decreasing monomorphism whose image does not include i𝑖i and σin:{0,,n+1}{0,,n}:subscriptsuperscript𝜎𝑛𝑖0𝑛10𝑛\sigma^{n}_{i}:\{0,\dots,n+1\}\to\{0,\dots,n\} is the non-decreasing epimorphism for which i𝑖i has two preimages. We shall write di,sisubscript𝑑𝑖subscript𝑠𝑖d_{i},s_{i} instead of din,sinsuperscriptsubscript𝑑𝑖𝑛superscriptsubscript𝑠𝑖𝑛d_{i}^{n},s_{i}^{n} when n𝑛n is clear from the context. Since the morphisms {σin,δin}i,nsubscriptsubscriptsuperscript𝜎𝑛𝑖subscriptsuperscript𝛿𝑛𝑖𝑖𝑛\{\sigma^{n}_{i},\delta^{n}_{i}\}_{i,n} generate 𝚫𝚫{\bm{\Delta}}, in order to specify a simplicial object Usubscript𝑈U_{\bullet} in 𝐗𝐗{\mathbf{X}}, it is enough to specify objects {Un}n0subscriptsubscript𝑈𝑛𝑛0\{U_{n}\}_{n\geq 0} and morphisms sin:UnUn+1:subscriptsuperscript𝑠𝑛𝑖subscript𝑈𝑛subscript𝑈𝑛1s^{n}_{i}:U_{n}\to U_{n+1}, din:UnUn1:subscriptsuperscript𝑑𝑛𝑖subscript𝑈𝑛subscript𝑈𝑛1d^{n}_{i}:U_{n}\to U_{n-1} for all 0in0𝑖𝑛0\leq i\leq n. Of course, the morphisms {sin,din}i,nsubscriptsubscriptsuperscript𝑠𝑛𝑖subscriptsuperscript𝑑𝑛𝑖𝑖𝑛\{s^{n}_{i},d^{n}_{i}\}_{i,n} have to satisfy certain relations, which can be found in [may_simplicial_1992], for instance.

For n0𝑛0n\geq 0, let 𝚫nsubscript𝚫absent𝑛{\bm{\Delta}}_{\leq n} denote the full subcategory of 𝚫𝚫{\bm{\Delta}} whose objects are {{0},,{0,,n}}00𝑛\{\{0\},\dots,\{0,\dots,n\}\}. The restriction functor UUn:Fun(𝚫op,𝐗)Fun(𝚫nop,𝐗):maps-tosubscript𝑈subscript𝑈absent𝑛Funsuperscript𝚫op𝐗Funsubscriptsuperscript𝚫opabsent𝑛𝐗U_{\bullet}\mapsto U_{\leq n}:\mathrm{Fun}({\bm{\Delta}}^{\text{op}},{\mathbf{X}})\to\mathrm{Fun}({\bm{\Delta}}^{\text{op}}_{\leq n},{\mathbf{X}}) admits a right adjoint called the n𝑛n-th coskeleton and denoted cosknsubscriptcosk𝑛{\mathrm{cosk}}_{n}. We also write coskn(U)subscriptcosk𝑛subscript𝑈{\mathrm{cosk}}_{n}(U_{\bullet}) for coskn(Un)subscriptcosk𝑛subscript𝑈absent𝑛{\mathrm{cosk}}_{n}(U_{\leq n}). The simplicial object Usubscript𝑈U_{\bullet} is called a hypercovering (of the terminal object) if U0subscript𝑈0U_{0}\to* is a covering and for all n0𝑛0n\geq 0, the map Un+1coskn(U)n+1subscript𝑈𝑛1subscriptcosk𝑛subscriptsubscript𝑈𝑛1U_{n+1}\to{\mathrm{cosk}}_{n}(U_{\bullet})_{n+1} induced by the adjunction is a covering. For example, when n=0𝑛0n=0, the latter conditions means that (d01,d11):U1U0×U0:superscriptsubscript𝑑01superscriptsubscript𝑑11subscript𝑈1subscript𝑈0subscript𝑈0(d_{0}^{1},d_{1}^{1}):U_{1}\to U_{0}\times U_{0} is a covering.

Hypercoverings form a category in the obvious manner, morphisms being natural transformations.

Example 2.3.1.

Let U𝑈U\to* be a morphism in 𝐗𝐗{\mathbf{X}}. Define Un=U××Usubscript𝑈𝑛𝑈𝑈U_{n}=U\times\dots\times U (n+1𝑛1n+1 times), let din:UnUn1:superscriptsubscript𝑑𝑖𝑛subscript𝑈𝑛subscript𝑈𝑛1d_{i}^{n}:U_{n}\to U_{n-1} be the projection omitting the i𝑖i-th copy of U𝑈U and let sin:UnUn+1:superscriptsubscript𝑠𝑖𝑛subscript𝑈𝑛subscript𝑈𝑛1s_{i}^{n}:U_{n}\to U_{n+1} be given by (u0,,un)(u0,,ui,ui,,un)maps-tosubscript𝑢0subscript𝑢𝑛subscript𝑢0subscript𝑢𝑖subscript𝑢𝑖subscript𝑢𝑛(u_{0},\dots,u_{n})\mapsto(u_{0},\dots,u_{i},u_{i},\dots,u_{n}) on sections. These data determine a simplicial object Usubscript𝑈U_{\bullet} which is a hypercovering if U𝑈U\to* is a covering. In this case, the map Un+1coskn(U)n+1subscript𝑈𝑛1subscriptcosk𝑛subscriptsubscript𝑈𝑛1U_{n+1}\to{\mathrm{cosk}}_{n}(U_{\bullet})_{n+1} is an isomorphism for all n𝑛n. The hypercovering Usubscript𝑈U_{\bullet} is called the Čech hypercovering associated to U𝑈U. If Usubscript𝑈U_{\bullet} is an arbitrary hypercovering, then cosk0(U)subscriptcosk0subscript𝑈{\mathrm{cosk}}_{0}(U_{\bullet}) is the Čech hypercovering associated to U0subscript𝑈0U_{0}.

The following lemma is fundamental.

Lemma 2.3.2 ([de_jong_stacks_2017, Lm. 24.7.3] or [artin_theorie_1972, Th. V.7.3.2]).

Let Usubscript𝑈U_{\bullet} be a hypercovering and let VUn𝑉subscript𝑈𝑛V\to U_{n} be a covering. Then there exists a hypercovering morphism UUsubscriptsuperscript𝑈subscript𝑈U^{\prime}_{\bullet}\to U_{\bullet} such that UnUnsubscriptsuperscript𝑈𝑛subscript𝑈𝑛{U^{\prime}_{n}\to U_{n}} factors through VUn𝑉subscript𝑈𝑛V\to U_{n}.

Let A𝐴A be an abelian group object of 𝐗𝐗\mathbf{X}. With any hypercovering Usubscript𝑈U_{\bullet} in 𝐗𝐗{\mathbf{X}} we associate a cochain complex C(U,A)superscript𝐶subscript𝑈𝐴C^{\bullet}(U_{\bullet},A) defined by Cn(U,A)=A(Un)superscript𝐶𝑛subscript𝑈𝐴𝐴subscript𝑈𝑛C^{n}(U_{\bullet},A)=A(U_{n}) for n0𝑛0n\geq 0 and Cn(U,A)=0superscript𝐶𝑛subscript𝑈𝐴0C^{n}(U_{\bullet},A)=0 otherwise. The coboundary map dn:Cn(U,A)Cn+1(U,A):superscript𝑑𝑛superscript𝐶𝑛subscript𝑈𝐴superscript𝐶𝑛1subscript𝑈𝐴d^{n}:C^{n}(U_{\bullet},A)\to C^{n+1}(U_{\bullet},A) is given by dn(a)=i=0n+1(1)idi(a)superscript𝑑𝑛𝑎superscriptsubscript𝑖0𝑛1superscript1𝑖subscriptsuperscript𝑑𝑖𝑎d^{n}(a)=\sum_{i=0}^{n+1}(-1)^{i}d^{*}_{i}(a), as usual; here di=(din+1):A(Un)A(Un+1):subscriptsuperscript𝑑𝑖superscriptsubscriptsuperscript𝑑𝑛1𝑖𝐴subscript𝑈𝑛𝐴subscript𝑈𝑛1d^{*}_{i}=(d^{n+1}_{i})^{*}:A(U_{n})\to A(U_{n+1}) is the map induced by din+1:Un+1Un:subscriptsuperscript𝑑𝑛1𝑖subscript𝑈𝑛1subscript𝑈𝑛d^{n+1}_{i}:U_{n+1}\to U_{n}. The cocycles, coboundaries, and cohomology groups of the complex are denoted Zn(U,A)superscript𝑍𝑛subscript𝑈𝐴Z^{n}(U_{\bullet},A), Bn(U,A)superscript𝐵𝑛subscript𝑈𝐴B^{n}(U_{\bullet},A) and Hn(U,A)superscriptH𝑛subscript𝑈𝐴\mathrm{H}^{n}(U_{\bullet},A). Any morphism of hypercoverings UUsubscriptsuperscript𝑈subscript𝑈U^{\prime}_{\bullet}\to U_{\bullet} induces a morphism Hn(U,A)Hn(U,A)superscriptH𝑛subscript𝑈𝐴superscriptH𝑛subscriptsuperscript𝑈𝐴\mathrm{H}^{n}(U_{\bullet},A)\to\mathrm{H}^{n}(U^{\prime}_{\bullet},A) in the obvious manner.

Theorem 2.3.3 (Verdier [artin_theorie_1972, Th. V.7.4.1]).

Let 𝐗𝐗\mathbf{X} be a topos and A𝐴A an abelian group object in 𝐗𝐗\mathbf{X}. The functors

AHn(𝐗,A)andAcolimUHn(U,A)formulae-sequencemaps-to𝐴superscriptH𝑛𝐗𝐴andmaps-to𝐴subscriptcolimsubscript𝑈superscriptH𝑛subscript𝑈𝐴A\mapsto\mathrm{H}^{n}({\mathbf{X}},A)\qquad\text{and}\qquad A\mapsto\operatornamewithlimits{colim}_{U_{\bullet}}\mathrm{H}^{n}(U_{\bullet},A)

from the category of abelian groups in 𝐗𝐗{\mathbf{X}} to the category of abelian groups are naturally isomorphic. Here, the colimit is taken over the category of hypercoverings.

Remark 2.3.4.

If we were to take the colimit in the theorem over the category of the Čech hypercoverings, then the result would be the Čech cohomology of A𝐴A. Consequently, the Čech cohomology and the derived-functor cohomology agree when every hypercovering admits a map from a Čech hypercovering. This is known to be the case when 𝐗=Sh(X)𝐗Sh𝑋{\mathbf{X}}=\text{\bf Sh}(X) for a paracompact Hausdorff topological space X𝑋X [godement_topologie_1973, Th. 5.10.1], or 𝐗=Sh(Xét)𝐗Shsubscript𝑋ét{\mathbf{X}}=\text{\bf Sh}(X_{\text{\'{e}t}}) for a noetherian scheme X𝑋X such that any finite subset of X𝑋X is contained in an open affine subscheme [artin_joins_1971, §4].

A short exact sequence of abelian groups 1AAA′′11superscript𝐴𝐴superscript𝐴′′11\to A^{\prime}\to A\to A^{\prime\prime}\to 1 in 𝐗𝐗{\mathbf{X}} gives rise to a long exact sequence of cohomology groups. By the second proof of [de_jong_stacks_2017, Tag 01H0], quoted as Theorem 2.3.3 here, the connecting homomorphism δn:Hn(A′′)Hn+1(A):superscript𝛿𝑛superscriptH𝑛superscript𝐴′′superscriptH𝑛1superscript𝐴\delta^{n}:\mathrm{H}^{n}(A^{\prime\prime})\to\mathrm{H}^{n+1}(A^{\prime}) can be described as follows: Let α′′Hn(A′′)superscript𝛼′′superscriptH𝑛superscript𝐴′′\alpha^{\prime\prime}\in\mathrm{H}^{n}(A^{\prime\prime}) be a cohomology class represented by a cocycle a′′Zn(U,A′′)superscript𝑎′′superscript𝑍𝑛subscript𝑈superscript𝐴′′a^{\prime\prime}\in Z^{n}(U_{\bullet},A^{\prime\prime}) for some hypercovering Usubscript𝑈U_{\bullet}. Since AA′′𝐴superscript𝐴′′A\to A^{\prime\prime} is an epimorphism, we may find a covering VUn𝑉subscript𝑈𝑛V\to U_{n} such that a′′A′′(Un)superscript𝑎′′superscript𝐴′′subscript𝑈𝑛a^{\prime\prime}\in A^{\prime\prime}(U_{n}) is the image of some aA(V)𝑎𝐴𝑉a\in A(V). By Lemma 2.3.2 there exists a morphism of hypercoverings VUsubscript𝑉subscript𝑈V_{\bullet}\to U_{\bullet} such that VnUnsubscript𝑉𝑛subscript𝑈𝑛V_{n}\to U_{n} factors through VUn𝑉subscript𝑈𝑛V\to U_{n}. We replace a𝑎a with its image in A(Vn)𝐴subscript𝑉𝑛A(V_{n}). One easily checks that the image of dn(a)Cn+1(V,A)superscript𝑑𝑛𝑎superscript𝐶𝑛1subscript𝑉𝐴d^{n}(a)\in C^{n+1}(V_{\bullet},A) in both Cn+1(V,A′′)superscript𝐶𝑛1subscript𝑉superscript𝐴′′C^{n+1}(V_{\bullet},A^{\prime\prime}) and Cn+2(V,A)superscript𝐶𝑛2subscript𝑉𝐴C^{n+2}(V_{\bullet},A) is 00, and hence dn(a)Zn+1(V,A)superscript𝑑𝑛𝑎superscript𝑍𝑛1subscript𝑉superscript𝐴d^{n}(a)\in Z^{n+1}(V_{\bullet},A^{\prime}). Now, δn(α′′)superscript𝛿𝑛superscript𝛼′′\delta^{n}(\alpha^{\prime\prime}) is the cohomology class determined by dn(a)Zn+1(V,A)superscript𝑑𝑛𝑎superscript𝑍𝑛1subscript𝑉superscript𝐴d^{n}(a)\in Z^{n+1}(V_{\bullet},A^{\prime}).

2.4. Cohomology of Non-Abelian Groups

For a group object G𝐺G of 𝐗𝐗\mathbf{X}, not necessary abelian, we define the pointed set H1(𝐗,G)superscriptH1𝐗𝐺\mathrm{H}^{1}(\mathbf{X},G) by hypercoverings. Given a hypercovering Usubscript𝑈U_{\bullet} in 𝐗𝐗{\mathbf{X}}, let Z1(U,G)superscript𝑍1subscript𝑈𝐺Z^{1}(U_{\bullet},G) be the set of elements gG(U1)𝑔𝐺subscript𝑈1g\in G(U_{1}) satisfying

(2) d2gd0gd1g1=1superscriptsubscript𝑑2𝑔superscriptsubscript𝑑0𝑔superscriptsubscript𝑑1superscript𝑔11d_{2}^{*}g\cdot d_{0}^{*}g\cdot d_{1}^{*}g^{-1}=1

in G(U2)𝐺subscript𝑈2G(U_{2}); here di=(di2):G(U1)G(U2):subscriptsuperscript𝑑𝑖superscriptsuperscriptsubscript𝑑𝑖2𝐺subscript𝑈1𝐺subscript𝑈2d^{*}_{i}=(d_{i}^{2})^{*}:G(U_{1})\to G(U_{2}) is induced by di2:U2U1:subscriptsuperscript𝑑2𝑖subscript𝑈2subscript𝑈1d^{2}_{i}:U_{2}\to U_{1}. Two elements g,gZ1(U,G)𝑔superscript𝑔superscript𝑍1subscript𝑈𝐺g,g^{\prime}\in Z^{1}(U_{\bullet},G) are said to be cohomologous, denoted ggsimilar-to𝑔superscript𝑔g\sim g^{\prime}, if there exists xG(U0)𝑥𝐺subscript𝑈0x\in G(U_{0}) such that g=d1xgd0x1𝑔superscriptsubscript𝑑1𝑥superscript𝑔superscriptsubscript𝑑0superscript𝑥1g=d_{1}^{*}x\cdot g^{\prime}\cdot d_{0}^{*}x^{-1}. We define the pointed set H1(U,G)superscriptH1subscript𝑈𝐺\mathrm{H}^{1}(U_{\bullet},G) to be Z1(U,G)/Z^{1}(U_{\bullet},G)/{\sim} with the equivalence class of 1G(U1)subscript1𝐺subscript𝑈11_{G(U_{1})} as a distinguished element. A morphism of hypercoverings UUsubscriptsuperscript𝑈subscript𝑈U^{\prime}_{\bullet}\to U_{\bullet} induces a morphism of pointed sets H1(U,G)H1(U,G)superscriptH1subscript𝑈𝐺superscriptH1subscriptsuperscript𝑈𝐺\mathrm{H}^{1}(U_{\bullet},G)\to\mathrm{H}^{1}(U^{\prime}_{\bullet},G). Now, following the literature, we define

H1(𝐗,G):=colimUH1(U,G),assignsuperscriptH1𝐗𝐺subscriptcolimsubscript𝑈superscriptH1subscript𝑈𝐺\mathrm{H}^{1}({\mathbf{X}},G):=\operatornamewithlimits{colim}_{U_{\bullet}}\mathrm{H}^{1}(U_{\bullet},G),

where the colimit is taken over the category of all hypercoverings in 𝐗𝐗{\mathbf{X}}. We note that some texts take the colimit over the category of Čech hypercoverings, see  Example 2.3.1, but this makes no difference thanks to the following lemma.

Lemma 2.4.1.

Let Usubscript𝑈U_{\bullet} be a hypercovering. Then the maps Z1(cosk0(U),G)Z1(U,G)superscript𝑍1subscriptcosk0subscript𝑈𝐺superscript𝑍1subscript𝑈𝐺Z^{1}({\mathrm{cosk}}_{0}(U_{\bullet}),G)\to Z^{1}(U_{\bullet},G) and H1(cosk0(U),G)H1(U,G)superscriptH1subscriptcosk0subscript𝑈𝐺superscriptH1subscript𝑈𝐺\mathrm{H}^{1}({\mathrm{cosk}}_{0}(U_{\bullet}),G)\to\mathrm{H}^{1}(U_{\bullet},G), induced by the canonical morphism Ucosk0(U)subscript𝑈subscriptcosk0subscript𝑈U_{\bullet}\to{\mathrm{cosk}}_{0}(U_{\bullet}), are isomorphisms.

Proof.

The proof shall require various facts about coskeleta. We refer the reader to [de_jong_stacks_2017, §14.19] or any equivalent source for proofs.

Recall from Example 2.3.1 that cosk0(U)subscriptcosk0subscript𝑈{\mathrm{cosk}}_{0}(U_{\bullet}) is nothing but the Čech hypercovering associated to U0subscript𝑈0U_{0}. Since Usubscript𝑈U_{\bullet} is a hypercovering, (d0,d1):U1cosk0(U)1=U0×U0:subscript𝑑0subscript𝑑1subscript𝑈1subscriptcosk0subscriptsubscript𝑈1subscript𝑈0subscript𝑈0(d_{0},d_{1}):U_{1}\to{\mathrm{cosk}}_{0}(U_{\bullet})_{1}=U_{0}\times U_{0} is a covering, and hence the induced map G(U0×U0)G(U1)𝐺subscript𝑈0subscript𝑈0𝐺subscript𝑈1G(U_{0}\times U_{0})\to G(U_{1}) is injective. Since the map Z1(cosk0(U),G)Z1(U,G)superscript𝑍1subscriptcosk0subscript𝑈𝐺superscript𝑍1subscript𝑈𝐺Z^{1}({\mathrm{cosk}}_{0}(U_{\bullet}),G)\to Z^{1}(U_{\bullet},G) is a restriction of the latter, it is also injective. This implies that if two cocycles in Z1(cosk0(U),G)superscript𝑍1subscriptcosk0subscript𝑈𝐺Z^{1}({\mathrm{cosk}}_{0}(U_{\bullet}),G) become cohomologous in Z1(U,G)superscript𝑍1subscript𝑈𝐺Z^{1}(U_{\bullet},G), then they are also cohomologous in Z1(cosk0(U),G)superscript𝑍1subscriptcosk0subscript𝑈𝐺Z^{1}({\mathrm{cosk}}_{0}(U_{\bullet}),G), so H1(cosk0(U),G)H1(U,G)superscriptH1subscriptcosk0subscript𝑈𝐺superscriptH1subscript𝑈𝐺\mathrm{H}^{1}({\mathrm{cosk}}_{0}(U_{\bullet}),G)\to\mathrm{H}^{1}(U_{\bullet},G) is injective. It is therefore enough to show that Z1(cosk0(U),G)Z1(U,G)superscript𝑍1subscriptcosk0subscript𝑈𝐺superscript𝑍1subscript𝑈𝐺Z^{1}({\mathrm{cosk}}_{0}(U_{\bullet}),G)\to Z^{1}(U_{\bullet},G) is surjective.

We first observe that the canonical map Z1(cosk1(U),G)Z1(U,G)superscript𝑍1subscriptcosk1subscript𝑈𝐺superscript𝑍1subscript𝑈𝐺Z^{1}({\mathrm{cosk}}_{1}(U_{\bullet}),G)\to Z^{1}(U_{\bullet},G) is an isomorphism. This follows from the fact that cosk1(U)1=U1subscriptcosk1subscriptsubscript𝑈1subscript𝑈1{\mathrm{cosk}}_{1}(U_{\bullet})_{1}=U_{1} and U2cosk1(U)2subscript𝑈2subscriptcosk1subscriptsubscript𝑈2U_{2}\to{\mathrm{cosk}}_{1}(U_{\bullet})_{2} is a covering, hence (2) is satisfied in G(U2)𝐺subscript𝑈2G(U_{2}) if and only if it is satisfied in G(cosk1(U)2)𝐺subscriptcosk1subscriptsubscript𝑈2G({\mathrm{cosk}}_{1}(U_{\bullet})_{2}). Since cosk0(cosk1(U))=cosk0(U)subscriptcosk0subscriptcosk1subscript𝑈subscriptcosk0subscript𝑈{\mathrm{cosk}}_{0}({\mathrm{cosk}}_{1}(U_{\bullet}))={\mathrm{cosk}}_{0}(U_{\bullet}), we may replace Usubscript𝑈U_{\bullet} with cosk1(U)subscriptcosk1subscript𝑈{\mathrm{cosk}}_{1}(U_{\bullet}). In this case, the construction of cosk1subscriptcosk1{\mathrm{cosk}}_{1} implies that U2subscript𝑈2U_{2} is characterized by

U2(X)={(e01,e02,e12,v0,v1,v2)U1(X)3×U0(X)3:\displaystyle U_{2}(X)=\{(e_{01},e_{02},e_{12},v_{0},v_{1},v_{2})\in U_{1}(X)^{3}\times U_{0}(X)^{3}\,:\,\, d0eij=vj,subscript𝑑0subscript𝑒𝑖𝑗subscript𝑣𝑗\displaystyle d_{0}e_{ij}=v_{j},
d1eij=vifor all legali,j}.\displaystyle d_{1}e_{ij}=v_{i}~{}\text{for all legal}~{}i,j\}\ .

for all objects X𝑋X of 𝐗𝐗\mathbf{X}. The maps d0,d1,d2:U2U1:subscript𝑑0subscript𝑑1subscript𝑑2subscript𝑈2subscript𝑈1d_{0},d_{1},d_{2}:U_{2}\to U_{1} are then given by taking the e12subscript𝑒12e_{12}-part, e02subscript𝑒02e_{02}-part, and e01subscript𝑒01e_{01}-part, respectively. Geometrically, U2subscript𝑈2U_{2} is the object of simplicial morphisms from the boundary of the 222-simplex to Usubscript𝑈U_{\bullet}.

Let gZ1(U,G)G(U1)𝑔superscript𝑍1subscript𝑈𝐺𝐺subscript𝑈1g\in Z^{1}(U_{\bullet},G)\subseteq G(U_{1}). We claim that g𝑔g descends along (d0,d1)subscript𝑑0subscript𝑑1(d_{0},d_{1}) to gG(U0×U0)superscript𝑔𝐺subscript𝑈0subscript𝑈0g^{\prime}\in G({U_{0}\times U_{0}}). Write V=U1×U0×U0U1𝑉subscriptsubscript𝑈0subscript𝑈0subscript𝑈1subscript𝑈1V=U_{1}\times_{U_{0}\times U_{0}}U_{1} and let π1subscript𝜋1\pi_{1}, π2subscript𝜋2\pi_{2} denote the first and second projections from V𝑉V onto U1subscript𝑈1U_{1}. We need to show that π1g=π2gsuperscriptsubscript𝜋1𝑔superscriptsubscript𝜋2𝑔\pi_{1}^{*}g=\pi_{2}^{*}g in G(V)𝐺𝑉G(V). For an object X𝑋X, the X𝑋X-sections of V𝑉V can be described by

V(X)={(e01,e01,v0,v1)X(U1)2×X(U0)2:\displaystyle V(X)=\{(e_{01},e^{\prime}_{01},v_{0},v_{1})\in X(U_{1})^{2}\times X(U_{0})^{2}\,:\, d0(e01)=d0(e01)=v1,subscript𝑑0subscript𝑒01subscript𝑑0subscriptsuperscript𝑒01subscript𝑣1\displaystyle d_{0}(e_{01})=d_{0}(e^{\prime}_{01})=v_{1},
d1(e01)=d1(e01)=v0}.\displaystyle d_{1}(e_{01})=d_{1}(e^{\prime}_{01})=v_{0}\}\ .

Define Ψ:VU2:Ψ𝑉subscript𝑈2\Psi:V\to U_{2} by

Ψ(e01,e01,v0,v1)=(e01,e01,s0v1,v0,v1,v1)Ψsubscript𝑒01subscriptsuperscript𝑒01subscript𝑣0subscript𝑣1subscript𝑒01subscriptsuperscript𝑒01subscript𝑠0subscript𝑣1subscript𝑣0subscript𝑣1subscript𝑣1\Psi(e_{01},e^{\prime}_{01},v_{0},v_{1})=(e_{01},e^{\prime}_{01},s_{0}v_{1},v_{0},v_{1},v_{1})

on sections. One readily checks that d0Ψ=s0d0π1=d0s1π1subscript𝑑0Ψsubscript𝑠0subscript𝑑0subscript𝜋1subscript𝑑0subscript𝑠1subscript𝜋1d_{0}\Psi=s_{0}d_{0}\pi_{1}=d_{0}s_{1}\pi_{1}, d1Ψ=π2subscript𝑑1Ψsubscript𝜋2d_{1}\Psi=\pi_{2} and d2Ψ=π1subscript𝑑2Ψsubscript𝜋1d_{2}\Psi=\pi_{1}. Now, applying Ψ:G(U2)G(V):superscriptΨ𝐺subscript𝑈2𝐺𝑉\Psi^{*}:G(U_{2})\to G(V) to (2), we arrive at the equation π1gπ1d0s1gπ2g1=1superscriptsubscript𝜋1𝑔superscriptsubscript𝜋1superscriptsubscript𝑑0superscriptsubscript𝑠1𝑔superscriptsubscript𝜋2superscript𝑔11\pi_{1}^{*}g\cdot\pi_{1}^{*}d_{0}^{*}s_{1}^{*}g\cdot\pi_{2}^{*}g^{-1}=1 in G(V)𝐺𝑉G(V), and applying s1:G(U2)G(U1):superscriptsubscript𝑠1𝐺subscript𝑈2𝐺subscript𝑈1s_{1}^{*}:G(U_{2})\to G(U_{1}) to (2), we find that gs1d0gg1=1𝑔superscriptsubscript𝑠1superscriptsubscript𝑑0𝑔superscript𝑔11g\cdot s_{1}^{*}d_{0}^{*}g\cdot g^{-1}=1 in G(U1)𝐺subscript𝑈1G(U_{1}). Both equations taken together imply that π1g=π2gsuperscriptsubscript𝜋1𝑔superscriptsubscript𝜋2𝑔\pi_{1}^{*}g=\pi_{2}^{*}g in G(V)𝐺𝑉G(V), hence our claim follows.

To finish the proof, it is enough to show that gsuperscript𝑔g^{\prime} is a 111-cocycle. This will follow from the fact that g𝑔g is a 111-cocycle if we show that the canonical map U2cosk0(U)2=U03subscript𝑈2subscriptcosk0subscriptsubscript𝑈2superscriptsubscript𝑈03U_{2}\to{\mathrm{cosk}}_{0}(U_{\bullet})_{2}=U_{0}^{3} is a covering. The latter map is given by (e01,e02,e12,v0,v1,v2)(v0,v1,v2)maps-tosubscript𝑒01subscript𝑒02subscript𝑒12subscript𝑣0subscript𝑣1subscript𝑣2subscript𝑣0subscript𝑣1subscript𝑣2(e_{01},e_{02},e_{12},v_{0},v_{1},v_{2})\mapsto(v_{0},v_{1},v_{2}) on sections. Since (d0,d1):U1U0:subscript𝑑0subscript𝑑1subscript𝑈1subscript𝑈0(d_{0},d_{1}):U_{1}\to U_{0} is a covering, for every pair of vertices v0,v1U0(X)subscript𝑣0subscript𝑣1subscript𝑈0𝑋v_{0},v_{1}\in U_{0}(X), there exists a covering XXsuperscript𝑋𝑋X^{\prime}\to X and an edge e01U1(X)subscript𝑒01subscript𝑈1superscript𝑋e_{01}\in U_{1}(X^{\prime}) satisfying d0e01=v1subscript𝑑0subscript𝑒01subscript𝑣1d_{0}e_{01}=v_{1}, d1e01=v0subscript𝑑1subscript𝑒01subscript𝑣0d_{1}e_{01}=v_{0}. This easily implies that U2U03subscript𝑈2superscriptsubscript𝑈03U_{2}\to U_{0}^{3} is locally surjective, finishing the proof. ∎

The following proposition summarizes the main properties of H1(𝐗,G)superscriptH1𝐗𝐺\mathrm{H}^{1}({\mathbf{X}},G). As before, we shall suppress 𝐗𝐗{\mathbf{X}}, writing H1(G)superscriptH1𝐺\mathrm{H}^{1}(G), when it is clear form the context.

Proposition 2.4.2.

Let 1GGG′′11superscript𝐺𝐺superscript𝐺′′11\to G^{\prime}\to G\to G^{\prime\prime}\to 1 be a short exact sequence of groups in 𝐗𝐗{\mathbf{X}}.

  1. (i)

    H1(G)=H1(𝐗,G)superscriptH1𝐺superscriptH1𝐗𝐺\mathrm{H}^{1}(G)=\mathrm{H}^{1}({\mathbf{X}},G) is naturally isomorphic to the set of isomorphism classes of G𝐺G-torsors; the distinguished element of H1(G)superscriptH1𝐺\mathrm{H}^{1}(G) corresponds to the isomorphism class of the trivial torsor.

  2. (ii)

    There is a long exact sequence of pointed sets

    1H0(G)H0(G)H0(G′′)δ1H1(G)H1(G)H1(G′′).1superscriptH0superscript𝐺superscriptH0𝐺superscriptH0superscript𝐺′′superscript𝛿1superscriptH1superscript𝐺superscriptH1𝐺superscriptH1superscript𝐺′′1\to\mathrm{H}^{0}(G^{\prime})\to\mathrm{H}^{0}(G)\to\mathrm{H}^{0}(G^{\prime\prime})\xrightarrow{\delta^{1}}\mathrm{H}^{1}(G^{\prime})\to\mathrm{H}^{1}(G)\to\mathrm{H}^{1}(G^{\prime\prime})\ .

    This exact sequence is functorial in 1GGG′′11superscript𝐺𝐺superscript𝐺′′11\to G^{\prime}\to G\to G^{\prime\prime}\to 1, i.e., a morphism from it to another short exact sequence of groups gives rise a morphism between the corresponding long exact sequences.

  3. (iii)

    When Gsuperscript𝐺G^{\prime} is central in G𝐺G, one can extend the exact sequence of (ii) with an additional morphism δ2:H1(G′′)H2(G):superscript𝛿2superscriptH1superscript𝐺′′superscriptH2superscript𝐺\delta^{2}:\mathrm{H}^{1}(G^{\prime\prime})\to\mathrm{H}^{2}(G^{\prime}), which is again functorial in 1GGG′′11superscript𝐺𝐺superscript𝐺′′11\to G^{\prime}\to G\to G^{\prime\prime}\to 1.

  4. (iv)

    When G,G,G′′superscript𝐺𝐺superscript𝐺′′G^{\prime},G,G^{\prime\prime} are abelian, the exact sequence of (iii) is canonically isomorphic to the truncation of the usual long exact sequence of cohomology groups associated to 1GGG′′11superscript𝐺𝐺superscript𝐺′′11\to G^{\prime}\to G\to G^{\prime\prime}\to 1. In particular, H1(G)superscriptH1𝐺\mathrm{H}^{1}(G) defined here is naturally isomorphic to H1(G)superscriptH1𝐺\mathrm{H}^{1}(G) defined in 2.3 and regarded as a pointed set with distinguished element 111.

The proposition is well known, but Giraud [giraud_cohomologie_1971, IV, 4.2.7.4, 4.2.10] is the only source we are aware of that treats all parts in the generality that we require. Since the treatment in [giraud_cohomologie_1971] is somewhat obscure, and since we shall need the definition of the maps δ1superscript𝛿1\delta^{1} and δ2superscript𝛿2\delta^{2} in the sequel, we include an outline of the proof here. Note that it is easier to prove (i) using the definition of H1(G)superscriptH1𝐺\mathrm{H}^{1}(G) via Čech hypercoverings, while it is easier to prove (iii) using the definition of H1(G)superscriptH1𝐺\mathrm{H}^{1}(G) via arbitrary hypercoverings, and these definitions are equivalent thanks to Lemma 2.4.1.

Proof (sketch).
  1. (i)

    Let P𝑃P be a G𝐺G-torsor. Choose a covering U0𝐗subscript𝑈0subscript𝐗U_{0}\to*_{\mathbf{X}} such that P(U0)𝑃subscript𝑈0P(U_{0})\neq\emptyset and fix some xP(U0)𝑥𝑃subscript𝑈0x\in P(U_{0}). Form the Čech hypercovering Usubscript𝑈U_{\bullet} associated to U0subscript𝑈0U_{0}. Then there exists a unique gG(U1)𝑔𝐺subscript𝑈1g\in G(U_{1}) such that d1xg=d0xsuperscriptsubscript𝑑1𝑥𝑔superscriptsubscript𝑑0𝑥d_{1}^{*}x\cdot g=d_{0}^{*}x in P(U1)𝑃subscript𝑈1P(U_{1}). We leave it to the reader to check that gZ1(U,G)𝑔superscript𝑍1subscript𝑈𝐺g\in Z^{1}(U_{\bullet},G) and the construction Pgmaps-to𝑃𝑔P\mapsto g induces a well-defined map from the isomorphism classes of Tors(G)Tors𝐺\text{\bf Tors}(G) to H1(G)superscriptH1𝐺\mathrm{H}^{1}(G) taking the trivial G𝐺G-torsor to the special element of H1(G)superscriptH1𝐺\mathrm{H}^{1}(G).

    In the other direction, let αH1(G)𝛼superscriptH1𝐺\alpha\in\mathrm{H}^{1}(G). By Lemma 2.4.1, α𝛼\alpha is represented by some gZ1(U,G)𝑔superscript𝑍1subscript𝑈𝐺g\in Z^{1}(U_{\bullet},G) where Usubscript𝑈U_{\bullet} is a Čech hypercovering. Define P𝑃P to be the object of 𝐗𝐗{\mathbf{X}} characterized by P(V)={xG(U0×V):gVd0x=d1x}𝑃𝑉conditional-set𝑥𝐺subscript𝑈0𝑉subscript𝑔𝑉superscriptsubscript𝑑0𝑥superscriptsubscript𝑑1𝑥P(V)=\{x\in G(U_{0}\times V)\,:\,g_{V}\cdot d_{0}^{*}x=d_{1}^{*}x\}; here, di:G(U0×V)G(U1×V):superscriptsubscript𝑑𝑖𝐺subscript𝑈0𝑉𝐺subscript𝑈1𝑉d_{i}^{*}:G(U_{0}\times V)\to G(U_{1}\times V) is induced by di×id:U1×VU0×V:subscript𝑑𝑖idsubscript𝑈1𝑉subscript𝑈0𝑉d_{i}\times\mathrm{id}:U_{1}\times V\to U_{0}\times V. There is a right G𝐺G-action on P𝑃P given by (x,h)xhU0maps-to𝑥𝑥subscriptsubscript𝑈0(x,h)\mapsto x\cdot h_{U_{0}} on sections. We leave it to the reader to check that P𝑃P is indeed a G𝐺G-torsor, and the assignment αPmaps-to𝛼𝑃\alpha\mapsto P defines an inverse to the map of the previous paragraph. In particular, note that P𝐗𝑃subscript𝐗P\to*_{\mathbf{X}} is a covering because gG(U0)𝑔𝐺subscript𝑈0g\in G(U_{0}); use the fact that Usubscript𝑈U_{\bullet} is a Čech hypercovering.

  2. (ii)

    Define δ1:H0(G′′)H1(G):superscript𝛿1superscriptH0superscript𝐺′′superscriptH1superscript𝐺\delta^{1}:\mathrm{H}^{0}(G^{\prime\prime})\to\mathrm{H}^{1}(G^{\prime}) as follows: Let g′′H0(G′′)superscript𝑔′′superscriptH0superscript𝐺′′g^{\prime\prime}\in\mathrm{H}^{0}(G^{\prime\prime}). There is a covering U0subscript𝑈0U_{0}\to* such that g′′superscript𝑔′′g^{\prime\prime} lifts to some gG(U0)𝑔𝐺subscript𝑈0g\in G(U_{0}). One easily checks that g:=d0gd1g1G(U1)assignsuperscript𝑔superscriptsubscript𝑑0𝑔superscriptsubscript𝑑1superscript𝑔1𝐺subscript𝑈1g^{\prime}:=d_{0}^{*}g\cdot d_{1}^{*}g^{-1}\in G(U_{1}) lies in Z1(G,U)superscript𝑍1superscript𝐺subscript𝑈Z^{1}(G^{\prime},U_{\bullet}) where Usubscript𝑈U_{\bullet} is the Čech hypercovering associated to U0subscript𝑈0U_{0}. We define δ1g′′superscript𝛿1superscript𝑔′′\delta^{1}g^{\prime\prime} to be the cohomology class represented by gsuperscript𝑔g^{\prime}, and leave it to the reader to check that this is well-defined. All other maps in the sequence are defined in the obvious manner and the exactness is easy to check.

  3. (iii)

    Define δ2:H1(G′′)H2(G):superscript𝛿2superscriptH1superscript𝐺′′superscriptH2superscript𝐺\delta^{2}:\mathrm{H}^{1}(G^{\prime\prime})\to\mathrm{H}^{2}(G^{\prime}) as follows: Let αH1(G′′)𝛼superscriptH1superscript𝐺′′\alpha\in\mathrm{H}^{1}(G^{\prime\prime}) be represented by g′′Z1(U,G′′)superscript𝑔′′superscript𝑍1subscript𝑈superscript𝐺′′g^{\prime\prime}\in Z^{1}(U_{\bullet},G^{\prime\prime}) where Usubscript𝑈U_{\bullet} is a hypercovering. There is a covering VU1𝑉subscript𝑈1V\to U_{1} such that g′′superscript𝑔′′g^{\prime\prime} lifts to some gG(V)𝑔𝐺𝑉g\in G(V). By Lemma 2.3.2, there is a morphism of hypercoverings UUsubscriptsuperscript𝑈subscript𝑈U^{\prime}_{\bullet}\to U_{\bullet} such that U1U1subscriptsuperscript𝑈1subscript𝑈1U^{\prime}_{1}\to U_{1} factors through VU1𝑉subscript𝑈1V\to U_{1}. We replace g𝑔g with its image in G(U1)𝐺subscriptsuperscript𝑈1G(U^{\prime}_{1}). Let g:=d2gd0gd1g1G(U1)assignsuperscript𝑔superscriptsubscript𝑑2𝑔superscriptsubscript𝑑0𝑔superscriptsubscript𝑑1superscript𝑔1𝐺subscriptsuperscript𝑈1g^{\prime}:=d_{2}^{*}g\cdot d_{0}^{*}g\cdot d_{1}^{*}g^{-1}\in G(U^{\prime}_{1}). It is easy to check that gsuperscript𝑔g^{\prime} lies in G(U1)superscript𝐺subscriptsuperscript𝑈1G^{\prime}(U^{\prime}_{1}) and defines a 222-cocycle of Gsuperscript𝐺G^{\prime} relative to Usubscriptsuperscript𝑈U^{\prime}_{\bullet}. We define δ2αsuperscript𝛿2𝛼\delta^{2}\alpha to be the cohomology class represented by gZ2(U,G)superscript𝑔superscript𝑍2subscriptsuperscript𝑈superscript𝐺g^{\prime}\in Z^{2}(U^{\prime}_{\bullet},G^{\prime}), and leave it to the reader to check that this is well-defined. The exactness of the sequence at H1(G′′)superscriptH1superscript𝐺′′\mathrm{H}^{1}(G^{\prime\prime}) is straightforward to check.

  4. (iv)

    Verdier’s Theorem gives rise to an obvious isomorphism between H1(G)superscriptH1𝐺\mathrm{H}^{1}(G) as defined here and H1(G)superscriptH1𝐺\mathrm{H}^{1}(G) as defined in 2.3. It is immediate from the definitions that this isomorphism also induces an isomorphism between the long exact sequences; see the proofs of (ii), (iii) and the comment at the end of 2.3. ∎

2.5. Azumaya Algebras

Let R𝑅R be a ring object of 𝐗𝐗\mathbf{X} and let n𝑛n be a positive integer. Recall that an Azumaya R𝑅R-algebra of degree n𝑛n is an R𝑅R-algebra A𝐴A in 𝐗𝐗\mathbf{X} that is locally isomorphic to Mn×n(R)subscriptM𝑛𝑛𝑅\mathrm{M}_{n\times n}(R), i.e., there exists a covering U𝑈U\to* such that AUMn×n(RU)subscript𝐴𝑈subscriptM𝑛𝑛subscript𝑅𝑈A_{U}\cong\mathrm{M}_{n\times n}(R_{U}) as RUsubscript𝑅𝑈R_{U}-algebras. The Azumaya R𝑅R-algebras of degree n𝑛n together with R𝑅R-algebra isomorphisms form a category which we denote by

Azn(𝐗,R).subscriptAz𝑛𝐗𝑅{\text{\bf Az}}_{n}({\mathbf{X}},R)\ .

If Asuperscript𝐴A^{\prime} is another Azumaya R𝑅R-algebra, we let omR-alg(A,A)𝑜subscript𝑚R-alg𝐴superscript𝐴\mathcal{H}\!om_{\text{$R$-alg}}(A,A^{\prime}) denote the subobject of the internal mapping object (A)Asuperscriptsuperscript𝐴𝐴(A^{\prime})^{A} of 𝐗𝐗\mathbf{X} consisting of R𝑅R-algebra homomorphisms. We define the group object 𝒜utR-alg(A)𝒜𝑢subscript𝑡R-alg𝐴\mathcal{A}ut_{\textrm{$R$-alg}}(A) similarly.

Remark 2.5.1.

We have defined here Azumaya algebras of constant degree only. When 𝐗𝐗{\mathbf{X}} is connected, these are all the Azumaya algebras, but in general, one has to allow the degree n𝑛n to take values in the global sections of the sheaf \mathbb{N} of positive integers on 𝐗𝐗\mathbf{X}. For any such n𝑛n one can define Mn×n(R)subscriptM𝑛𝑛𝑅\mathrm{M}_{n\times n}(R) and the definition of Azumaya algebras of degree n𝑛n extends verbatim. We ignore this technicality, both for the sake of simplicity, and also since it is unnecessary for connected topoi, which are the topoi of interest to us.

Let 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}} be a ring object in 𝐗𝐗{\mathbf{X}}. Recall that 𝐗𝐗{\mathbf{X}} is locally ringed by 𝒪𝐗subscript𝒪𝐗{\mathcal{O}_{\mathbf{X}}}, or 𝒪𝐗subscript𝒪𝐗{\mathcal{O}_{\mathbf{X}}} is a local ring object in 𝐗𝐗{\mathbf{X}}, if for any object U𝑈U in 𝐗𝐗{\mathbf{X}} and {ri}iI𝒪𝐗(U)subscriptsubscript𝑟𝑖𝑖𝐼subscript𝒪𝐗𝑈\{r_{i}\}_{i\in I}\subseteq{\mathcal{O}_{\mathbf{X}}}(U) with 𝒪𝐗(U)=iri𝒪𝐗(U)subscript𝒪𝐗𝑈subscript𝑖subscript𝑟𝑖subscript𝒪𝐗𝑈{\mathcal{O}_{\mathbf{X}}}(U)=\sum_{i}r_{i}{\mathcal{O}_{\mathbf{X}}}(U), there exists a covering {UiU}iIsubscriptsubscript𝑈𝑖𝑈𝑖𝐼\{U_{i}\to U\}_{i\in I} such that ri𝒪𝐗(Ui)×subscript𝑟𝑖subscript𝒪𝐗superscriptsubscript𝑈𝑖r_{i}\in{{\mathcal{O}_{\mathbf{X}}}(U_{i})^{\times}} for all iI𝑖𝐼i\in I. In fact, one can take Ui=𝐗subscript𝑈𝑖subscript𝐗U_{i}=\emptyset_{{\mathbf{X}}} for almost all i𝑖i. We remark that the condition should also hold when I=𝐼I=\emptyset, which implies that 𝒪𝐗(U)subscript𝒪𝐗𝑈{\mathcal{O}}_{\mathbf{X}}(U) cannot be the zero ring when U𝐗𝑈subscript𝐗U\ncong\emptyset_{\mathbf{X}}. When 𝐗𝐗{\mathbf{X}} has enough points, the condition is equivalent to saying that for every point i:𝐩𝐭𝐗:𝑖𝐩𝐭𝐗i:\mathbf{pt}\to{\mathbf{X}}, the ring i𝒪𝐗superscript𝑖subscript𝒪𝐗i^{*}{\mathcal{O}_{\mathbf{X}}} is local (the zero ring is not considered local).

Suppose that 𝒪𝐗subscript𝒪𝐗{\mathcal{O}_{\mathbf{X}}} is a local ring object. Then the group homomorphism GLn(𝒪𝐗)𝒜ut𝒪𝐗-alg(Mn×n(𝒪𝐗))subscriptGL𝑛subscript𝒪𝐗𝒜𝑢subscript𝑡𝒪𝐗-algsubscriptM𝑛𝑛subscript𝒪𝐗\operatorname{GL}_{n}({\mathcal{O}}_{\mathbf{X}})\to\mathcal{A}ut_{\text{${\mathcal{O}_{\mathbf{X}}}$-alg}}(\mathrm{M}_{n\times n}({\mathcal{O}_{\mathbf{X}}})) given by a[xaxa1]maps-to𝑎delimited-[]maps-to𝑥𝑎𝑥superscript𝑎1a\mapsto[x\mapsto axa^{-1}] on sections is surjective, [giraud_cohomologie_1971, V.§4]. This induces an isomorphism PGLn(𝒪𝐗):=GLn(𝒪𝐗)/𝒪𝐗×𝒜ut𝒪𝐗-alg(Mn×n(𝒪𝐗))assignsubscriptPGL𝑛subscript𝒪𝐗subscriptGL𝑛subscript𝒪𝐗superscriptsubscript𝒪𝐗𝒜𝑢subscript𝑡𝒪𝐗-algsubscriptM𝑛𝑛subscript𝒪𝐗\operatorname{PGL}_{n}({\mathcal{O}_{\mathbf{X}}}):=\operatorname{GL}_{n}({\mathcal{O}_{\mathbf{X}}})/{\mathcal{O}_{\mathbf{X}}^{\times}}\to\mathcal{A}ut_{\text{${\mathcal{O}_{\mathbf{X}}}$-alg}}(\mathrm{M}_{n\times n}({\mathcal{O}_{\mathbf{X}}})), which will be used to freely identify the source and target in the sequel. The following proposition is well established, again see [giraud_cohomologie_1971, V.§4].

Proposition 2.5.2.

If 𝒪𝐗subscript𝒪𝐗{\mathcal{O}_{\mathbf{X}}} is a local ring object, then there is an equivalence of categories

Tors(𝐗,PGLn(𝒪𝐗))Azn(𝐗,𝒪𝐗)Tors𝐗subscriptPGL𝑛subscript𝒪𝐗similar-tosubscriptAz𝑛𝐗subscript𝒪𝐗\text{\bf Tors}({\mathbf{X}},\operatorname{PGL}_{n}({\mathcal{O}_{\mathbf{X}}}))\overset{\sim}{\longrightarrow}{\text{\bf Az}}_{n}({\mathbf{X}},{\mathcal{O}_{\mathbf{X}}})

given by the functors PP×PGLn(𝒪𝐗)Mn×n(𝒪𝐗)maps-to𝑃superscriptsubscriptPGL𝑛subscript𝒪𝐗𝑃subscriptM𝑛𝑛subscript𝒪𝐗P\mapsto P\times^{\operatorname{PGL}_{n}({\mathcal{O}_{\mathbf{X}}})}\mathrm{M}_{n\times n}({\mathcal{O}_{\mathbf{X}}}) and Aom𝒪𝐗-alg(Mn×n(𝒪𝐗),A)maps-to𝐴𝑜subscript𝑚𝒪𝐗-algsubscriptM𝑛𝑛subscript𝒪𝐗𝐴A\mapsto\mathcal{H}\!om_{\text{${\mathcal{O}_{\mathbf{X}}}$-alg}}(\mathrm{M}_{n\times n}({\mathcal{O}_{\mathbf{X}}}),A).

The proposition holds for any ring object 𝒪𝐗subscript𝒪𝐗{\mathcal{O}_{\mathbf{X}}} of 𝐗𝐗\mathbf{X} if one replaces the group object PGLn(𝒪𝐗)subscriptPGL𝑛subscript𝒪𝐗\operatorname{PGL}_{n}({\mathcal{O}_{\mathbf{X}}}) with 𝒜ut𝒪𝐗-alg(Mn×n(𝒪𝐗))𝒜𝑢subscript𝑡𝒪𝐗-algsubscriptM𝑛𝑛subscript𝒪𝐗\mathcal{A}ut_{\textrm{${\mathcal{O}_{\mathbf{X}}}$-alg}}(\mathrm{M}_{n\times n}({\mathcal{O}_{\mathbf{X}}})).

We continue to assume that 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{{\mathbf{X}}} is a local ring object. By Proposition 2.4.2(iii), the short exact sequence 1𝒪𝐗×GLn(𝒪𝐗)PGLn(𝒪𝐗)11superscriptsubscript𝒪𝐗subscriptGL𝑛subscript𝒪𝐗subscriptPGL𝑛subscript𝒪𝐗11\to{\mathcal{O}_{\mathbf{X}}^{\times}}\to\operatorname{GL}_{n}({\mathcal{O}_{\mathbf{X}}})\to\operatorname{PGL}_{n}({\mathcal{O}_{\mathbf{X}}})\to 1 gives rise to a pointed set map H1(PGLn(𝒪𝐗))H2(𝒪𝐗×)superscriptH1subscriptPGL𝑛subscript𝒪𝐗superscriptH2superscriptsubscript𝒪𝐗\mathrm{H}^{1}(\operatorname{PGL}_{n}({\mathcal{O}_{\mathbf{X}}}))\to\mathrm{H}^{2}({\mathcal{O}_{\mathbf{X}}^{\times}}); here, H()=H(𝐗,)superscriptHsuperscriptH𝐗\mathrm{H}^{*}(-)=\mathrm{H}^{*}({\mathbf{X}},-). As usual, the Brauer group of 𝒪𝐗subscript𝒪𝐗{\mathcal{O}_{\mathbf{X}}} is

Br(𝐗,𝒪𝐗)=Br(𝒪𝐗):=nim(H1(PGLn(𝒪𝐗))H2(𝒪𝐗×))Br𝐗subscript𝒪𝐗Brsubscript𝒪𝐗assignsubscript𝑛imsuperscriptH1subscriptPGL𝑛subscript𝒪𝐗superscriptH2superscriptsubscript𝒪𝐗\operatorname{Br}({\mathbf{X}},{\mathcal{O}_{\mathbf{X}}})=\operatorname{Br}({\mathcal{O}_{\mathbf{X}}}):=\bigcup_{n\in\mathbb{N}}\operatorname{im}\left(\mathrm{H}^{1}(\operatorname{PGL}_{n}({\mathcal{O}_{\mathbf{X}}}))\to\mathrm{H}^{2}({\mathcal{O}_{\mathbf{X}}^{\times}})\right)

the addition being that inherited from the group H2(𝒪𝐗×)superscriptH2superscriptsubscript𝒪𝐗\mathrm{H}^{2}({\mathcal{O}_{\mathbf{X}}^{\times}}). Since Azumaya 𝒪Xsubscript𝒪𝑋\mathcal{O}_{X}-algebras correspond to PGLn(𝒪X)subscriptPGL𝑛subscript𝒪𝑋\operatorname{PGL}_{n}(\mathcal{O}_{X})-torsors, which are in turn classified by H1(𝐗,PGLn(𝒪X))superscriptH1𝐗subscriptPGL𝑛subscript𝒪𝑋\mathrm{H}^{1}(\mathbf{X},\operatorname{PGL}_{n}(\mathcal{O}_{X})), any Azumaya 𝒪𝐗subscript𝒪𝐗{\mathcal{O}_{\mathbf{X}}}-algebra A𝐴A gives rise to an element in Br(𝒪𝐗)Brsubscript𝒪𝐗\operatorname{Br}({\mathcal{O}_{\mathbf{X}}}), denoted [A]delimited-[]𝐴[A] and called the Brauer class of A𝐴A. By writing A[A]superscript𝐴delimited-[]𝐴A^{\prime}\in[A] or saying that Asuperscript𝐴A^{\prime} is Brauer equivalent to A𝐴A, we mean that Asuperscript𝐴A^{\prime} is an Azumaya 𝒪𝐗subscript𝒪𝐗{\mathcal{O}_{\mathbf{X}}}-algebra with [A]=[A]delimited-[]superscript𝐴delimited-[]𝐴[A^{\prime}]=[A]. For more details, see [grothendieck_groupe_1968-1] or [giraud_cohomologie_1971, Chap. V, §4].

Example 2.5.3.

Let X𝑋X be a topological space, let 𝐗=Sh(X)𝐗Sh𝑋{\mathbf{X}}=\text{\bf Sh}(X) and let 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}} be the sheaf of continuous functions from X𝑋X to \mathbb{C}, denoted 𝒞(X,)𝒞𝑋{\mathcal{C}}(X,\mathbb{C}). Then Azumaya 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}}-algebra are topological Azumaya algebras over X𝑋X as studied in [antieau_period-index_2014].

Example 2.5.4.

An Azumaya algebra A𝐴A of degree n𝑛n over a scheme X𝑋X is a sheaf of 𝒪Xsubscript𝒪𝑋{\mathcal{O}}_{X}-algebras that is locally, in the étale topology, isomorphic as an 𝒪Xsubscript𝒪𝑋\mathcal{O}_{X}-algebra to Matn×n(𝒪X)subscriptMat𝑛𝑛subscript𝒪𝑋\operatorname{Mat}_{n\times n}(\mathcal{O}_{X}), [grothendieck_groupe_1968-1, Para. 1.2]

Example 2.5.5.

Let R𝑅R be a ring. An Azumaya R𝑅R-algebra of degree n𝑛n is an R𝑅R-algebra A𝐴A for which there exists a faithfully flat étale R𝑅R-algebra Rsuperscript𝑅R^{\prime} such that ARRMn×n(R)subscripttensor-product𝑅𝐴superscript𝑅subscriptM𝑛𝑛superscript𝑅A\otimes_{R}R^{\prime}\cong\mathrm{M}_{n\times n}(R^{\prime}) as Rsuperscript𝑅R^{\prime}-algebras. This is equivalent to the definition of Example 2.5.4 in the case where X=SpecR𝑋Spec𝑅X=\operatorname{Spec}R by [grothendieck_groupe_1968-1, Th. 5.1, Cor. 5.2]. Consult [knus_quadratic_1991, III.§5] for other equivalent definitions and cf. Remark 2.5.1.

3. Rings with Involution

In this section we collect a number of results regarding involutions of rings that will be needed later in the paper. The main result is Theorem 3.3.8, which gives the structure of those rings with involution (R,λ)𝑅𝜆(R,\lambda) for which the fixed ring of λ𝜆\lambda is local. It is shown that in this case, R𝑅R is a local ring in its own right, or R𝑅R is a quadratic étale algebra over the fixed ring of λ𝜆\lambda. In particular, the ring R𝑅R is semilocal.

Throughout, involutions will be written exponentially and the Jacobson radical of a ring R𝑅R will be denoted Jac(R)Jac𝑅\mathrm{Jac}(R).

3.1. Quadratic Étale Algebras

Definition 3.1.1.

Let S𝑆S be a ring. A commutative S𝑆S-algebra R𝑅R is said to be finite étale of rank n𝑛n if R𝑅R is a locally free S𝑆S-module of rank n𝑛n, and the multiplication map μ:RSRR:𝜇subscripttensor-product𝑆𝑅𝑅𝑅\mu:R\otimes_{S}R\to R may be split as a morphism of RSRsubscripttensor-product𝑆𝑅𝑅R\otimes_{S}R-modules, where R𝑅R is regarded as an RSRsubscripttensor-product𝑆𝑅𝑅R\otimes_{S}R-algebra via μ𝜇\mu. Finite étale S𝑆S-algebras of rank 222 will be called quadratic étale algebras.

Remark 3.1.2.

One common definition of étale for commutative S𝑆S-algebras is that R𝑅R should be flat over S𝑆S, of finite presentation as an S𝑆S-algebra, and unramified in the sense that ΩR/SsubscriptΩ𝑅𝑆\Omega_{R/S}, the module of Kähler differentials, vanishes. This is the definition in [grothendieck_elements_1967, Sec. 17.6] in the affine case. Our finite étale algebras of rank n𝑛n are precisely the étale algebras that are locally free of rank n𝑛n.

Indeed, if R𝑅R is locally free of rank n𝑛n over S𝑆S, then R𝑅R is also finitely presented and flat as an S𝑆S-module [de_jong_stacks_2017, Tag 00NX], and hence of finite presentation as an R𝑅R-algebra [grothendieck_elements_1964, Prop. 1.4.7]. Furthermore, μ:RSRR:𝜇subscripttensor-product𝑆𝑅𝑅𝑅\mu:R\otimes_{S}R\to R admits a splitting ψ𝜓\psi if and only if the RSRsubscripttensor-product𝑆𝑅𝑅R\otimes_{S}R-ideal kerμkernel𝜇\ker\mu is generated by an idempotent, namely 1ψ(1R)1𝜓subscript1𝑅1-\psi(1_{R}). For finitely generated S𝑆S-algebras R𝑅R, the existence of such an idempotent is equivalent to saying R𝑅R is unramified over S𝑆S by [de_jong_stacks_2017, Tag 02FL].

Example 3.1.3.

Let fS[x]𝑓𝑆delimited-[]𝑥f\in S[x] be a monic polynomial of degree n𝑛n. It is well known that S[x]/(f)𝑆delimited-[]𝑥𝑓S[x]/(f) is a finite étale S𝑆S-algebra of rank n𝑛n if and only if the discriminant of f𝑓f is invertible in S𝑆S. In particular, S[x]/(x2+αx+β)𝑆delimited-[]𝑥superscript𝑥2𝛼𝑥𝛽S[x]/(x^{2}+\alpha x+\beta) is a quadratic étale S𝑆S-algebra if and only if α24βS×superscript𝛼24𝛽superscript𝑆\alpha^{2}-4\beta\in{S^{\times}}.

Every quadratic étale S𝑆S-algebra R𝑅R admits a canonical S𝑆S-linear involution λ𝜆\lambda given by rλ=TrR/S(r)rsuperscript𝑟𝜆subscriptTr𝑅𝑆𝑟𝑟r^{\lambda}=\mathrm{Tr}_{R/S}(r)-r, see [knus_quadratic_1991, I.§1.3.6]. The fixed ring of λ𝜆\lambda is S𝑆S and λ𝜆\lambda is the only S𝑆S-automorphism of R𝑅R with this property. Moreover, when S𝑆S is connected, it is the only non-trivial S𝑆S-automorphism of R𝑅R.

Proposition 3.1.4.

Let S𝑆S be a local ring with maximal ideal 𝔪𝔪{\mathfrak{m}} and residue field k𝑘k, let R𝑅R be a quadratic étale S𝑆S-algebra, and let λ𝜆\lambda be the unique non-trivial S𝑆S-automorphism of R𝑅R. Then:

  1. (i)

    Jac(R)=R𝔪Jac𝑅𝑅𝔪\mathrm{Jac}(R)=R{\mathfrak{m}}

  2. (ii)

    R¯:=R/R𝔪assign¯𝑅𝑅𝑅𝔪\overline{R}:=R/R{\mathfrak{m}} is either a separable quadratic field extension of k𝑘k, or R¯k×k¯𝑅𝑘𝑘\overline{R}\cong k\times k. The automorphism that λ𝜆\lambda induces on R¯¯𝑅\overline{R} is the unique non-trivial k𝑘k-automorphism of R¯¯𝑅\overline{R}.

  3. (iii)

    (“Hilbert 90”) For every rR×𝑟superscript𝑅r\in{R^{\times}} with rλr=1superscript𝑟𝜆𝑟1r^{\lambda}r=1, there exists aR×𝑎superscript𝑅a\in{R^{\times}} such that r=a1aλ𝑟superscript𝑎1superscript𝑎𝜆r=a^{-1}a^{\lambda}.

Proof.

It is clear that R¯¯𝑅\overline{R} is a quadratic étale k𝑘k-algebra, and hence a product of separable field extensions of k𝑘k [demeyer_separable_1971, Th. II.2.5]. This implies the first assertion of (ii) as well as Jac(R)R𝔪Jac𝑅𝑅𝔪\mathrm{Jac}(R)\subseteq R{\mathfrak{m}}. The inclusion R𝔪Jac(R)𝑅𝔪Jac𝑅R{\mathfrak{m}}\subseteq\mathrm{Jac}(R) holds because R𝑅R is a finite S𝑆S-module [reiner_maximal_1975-1, Th. 6.15], so we have proved (i). The last assertion of (ii) follows from the fact that λ𝜆\lambda is given by xTrR/S(x)xmaps-to𝑥subscriptTr𝑅𝑆𝑥𝑥x\mapsto\mathrm{Tr}_{R/S}(x)-x.

Let r¯¯𝑟\overline{r} denote the image of rR𝑟𝑅r\in R in R¯¯𝑅\overline{R}. To prove (iii), we first claim that there is x¯R¯¯𝑥¯𝑅\overline{x}\in\overline{R} with x¯+x¯λr¯R¯×¯𝑥superscript¯𝑥𝜆¯𝑟superscript¯𝑅\overline{x}+\overline{x}^{\lambda}\overline{r}\in{\overline{R}^{\times}}. This is easy to see if R¯=k×k¯𝑅𝑘𝑘\overline{R}=k\times k. Otherwise, R¯¯𝑅\overline{R} is a field and such x¯¯𝑥\overline{x} exists unless x¯=x¯λr¯¯𝑥superscript¯𝑥𝜆¯𝑟\overline{x}=-\overline{x}^{\lambda}\overline{r} for all x¯R¯¯𝑥¯𝑅\overline{x}\in\overline{R}. The latter forces r¯=1¯𝑟1\overline{r}=-1 (take x¯=1¯𝑥1\overline{x}=1) and λR¯=idR¯subscript𝜆¯𝑅subscriptid¯𝑅\lambda_{\overline{R}}=\mathrm{id}_{\overline{R}}, which is impossible by (ii), so x¯¯𝑥\overline{x} exists.

Let xR𝑥𝑅x\in R be a lift of x¯¯𝑥\overline{x}. Then t:=x+xλrassign𝑡𝑥superscript𝑥𝜆𝑟t:=x+x^{\lambda}r is a lift of x¯+x¯λr¯¯𝑥superscript¯𝑥𝜆¯𝑟\overline{x}+\overline{x}^{{\lambda}}\overline{r}, which implies tR×𝑡superscript𝑅t\in{R^{\times}} by (i). Since rλ=r1superscript𝑟𝜆superscript𝑟1r^{\lambda}=r^{-1}, it is the case that tλr=tsuperscript𝑡𝜆𝑟𝑡t^{\lambda}r=t, and so r=aλa1𝑟superscript𝑎𝜆superscript𝑎1r=a^{\lambda}a^{-1} with a=t1𝑎superscript𝑡1a=t^{-1}. ∎

3.2. Quadratic Étale Algebras in Topoi

Our definition of a “quadratic étale algebra” extends directly to the case where S𝑆S is a local ring object in a topos 𝐗𝐗{\mathbf{X}}.

Definition 3.2.1.

Given a ring object S𝑆S in a topos 𝐗𝐗\mathbf{X}, we say an S𝑆S-algebra R𝑅R is a finite étale S𝑆S-algebra of rank n𝑛n if R𝑅R is a locally free S𝑆S-module of rank n𝑛n such that the multiplication map μ:RSRR:𝜇subscripttensor-product𝑆𝑅𝑅𝑅\mu:R\otimes_{S}R\to R may be split as a morphism of RSRsubscripttensor-product𝑆𝑅𝑅R\otimes_{S}R-algebras. Finite étale S𝑆S-algebras of rank 222 will be called quadratic étale algebras.

We alert the reader that if R𝑅R is a quadratic étale S𝑆S-algebra, then it is not true in general that R(U)𝑅𝑈R(U) is a quadratic étale S(U)𝑆𝑈S(U)-algebra for all objects U𝑈U of 𝐗𝐗{\mathbf{X}}. In fact R(U)𝑅𝑈R(U) may not be locally free of rank 222 over S(U)𝑆𝑈S(U). Rather, one can always find a covering VU𝑉𝑈V\to U such that R(V)𝑅𝑉R(V) is a quadratic étale S(V)𝑆𝑉S(V)-algebra; for instance, one may take any VU𝑉𝑈V\to U such that RVsubscript𝑅𝑉R_{V} is a free SVsubscript𝑆𝑉S_{V}-module of rank 222. We further note that in general there is no covering U𝑈U\to* such that RUSU×SUsubscript𝑅𝑈subscript𝑆𝑈subscript𝑆𝑈R_{U}\cong S_{U}\times S_{U} as SUsubscript𝑆𝑈S_{U}-algebras, e.g. let 𝐗=𝐩𝐭𝐗𝐩𝐭{\mathbf{X}}=\mathbf{pt} (the topos of sets) and take S𝑆S and R𝑅R to be \mathbb{Q} and [2]delimited-[]2\mathbb{Q}[\sqrt{2}] respectively. While μ:RSRR:𝜇subscripttensor-product𝑆𝑅𝑅𝑅\mu:R\otimes_{S}R\to R is easily seen to be split, there is no covering U𝑈U\to* such that RUSU×SUsubscript𝑅𝑈subscript𝑆𝑈subscript𝑆𝑈R_{U}\cong S_{U}\times S_{U}.

As one might expect, being a finite étale algebra of rank n𝑛n is a local property in that it may be tested on a covering.

Lemma 3.2.2.

Let S𝑆S be a ring in 𝐗𝐗{\mathbf{X}}, let R𝑅R be an S𝑆S-algebra and let U𝑈U\to* be a covering. Then R𝑅R is a finite étale S𝑆S-algebra of rank n𝑛n if and only if RUsubscript𝑅𝑈R_{U} is a finite étale SUsubscript𝑆𝑈S_{U}-algebra of rank n𝑛n in 𝐗/U𝐗𝑈{\mathbf{X}}/U.

Proof.

Write M=RSR𝑀subscripttensor-product𝑆𝑅𝑅M=R\otimes_{S}R. The only non-trivial thing to check is that if the multiplication map μU:MURU:subscript𝜇𝑈subscript𝑀𝑈subscript𝑅𝑈\mu_{U}:M_{U}\to R_{U} admits a splitting ψ:RUMU:𝜓subscript𝑅𝑈subscript𝑀𝑈\psi:R_{U}\to M_{U} in 𝐗/U𝐗𝑈{\mathbf{X}}/U, then so does μ:MR:𝜇𝑀𝑅\mu:M\to R in 𝐗𝐗{\mathbf{X}}. Let π1,π2:U×UU:subscript𝜋1subscript𝜋2𝑈𝑈𝑈\pi_{1},\pi_{2}:U\times U\to U denote the first and second projections, and let πiψ:RU×UMU×U:superscriptsubscript𝜋𝑖𝜓subscript𝑅𝑈𝑈subscript𝑀𝑈𝑈\pi_{i}^{*}\psi:R_{U\times U}\to M_{U\times U} denote the pullback of ψ𝜓\psi along πisubscript𝜋𝑖\pi_{i}. We claim that μU×U:MU×URU×U:subscript𝜇𝑈𝑈subscript𝑀𝑈𝑈subscript𝑅𝑈𝑈\mu_{U\times U}:M_{U\times U}\to R_{U\times U} admits at most one splitting. Provided this holds, we must have π1ψ=π2ψsuperscriptsubscript𝜋1𝜓superscriptsubscript𝜋2𝜓\pi_{1}^{*}\psi=\pi_{2}^{*}\psi and so ψ𝜓\psi descends to a map ψ0:RM:subscript𝜓0𝑅𝑀\psi_{0}:R\to M splitting μ𝜇\mu as required.

The claim can be verified on the level of sections, namely, it is enough to check that any ring surjection μ:AB:𝜇𝐴𝐵\mu:A\to B admits at most one A𝐴A-linear splitting. If ψ𝜓\psi is such a splitting and e=ψ(1B)𝑒𝜓subscript1𝐵e=\psi(1_{B}), then ψ(1Bα)=αe𝜓subscript1𝐵𝛼𝛼𝑒\psi(1_{B}\cdot\alpha)=\alpha e for all αA𝛼𝐴\alpha\in A, so ψ𝜓\psi is determined by the idempotent e𝑒e. It is easy to check that (1e)A=kerμ1𝑒𝐴kernel𝜇(1-e)A=\ker\mu and that e𝑒e is the only idempotent with this property, hence e𝑒e is determined by μ𝜇\mu. ∎

Example 3.2.3.

Let π:XY:𝜋𝑋𝑌\pi:X\to Y be a quadratic étale morphism of schemes. That is, π𝜋\pi is affine and Y𝑌Y can be covered by open affine subschemes {Ui}isubscriptsubscript𝑈𝑖𝑖\{U_{i}\}_{i} such that the ring map corresponding to π:π1(Ui)Ui:𝜋superscript𝜋1subscript𝑈𝑖subscript𝑈𝑖\pi:\pi^{-1}(U_{i})\to U_{i} is quadratic étale for all i𝑖i. Then π𝒪Xsubscript𝜋subscript𝒪𝑋\pi_{*}{\mathcal{O}}_{X} is a quadratic étale 𝒪Ysubscript𝒪𝑌{\mathcal{O}}_{Y}-algebra in both Sh(Yét)Shsubscript𝑌ét\text{\bf Sh}(Y_{\text{\'{e}t}}) and Sh(YZar)Shsubscript𝑌Zar\text{\bf Sh}(Y_{\operatorname{Zar}}); this can be checked using Lemma 3.2.2.

Example 3.2.4.

Let π:XY:𝜋𝑋𝑌\pi:X\to Y be a double covering of topological spaces and let 𝒞(X,)𝒞𝑋{\mathcal{C}}(X,\mathbb{C}) and 𝒞(Y,)𝒞𝑌{\mathcal{C}}(Y,\mathbb{C}) denote the sheaves of continuous \mathbb{C}-values functions on X𝑋X and Y𝑌Y, respectively. Then π𝒞(X,)subscript𝜋𝒞𝑋\pi_{*}{\mathcal{C}}(X,\mathbb{C}) is a quadratic étale 𝒞(Y,)𝒞𝑌{\mathcal{C}}(Y,\mathbb{C})-algebra in Sh(Y)Sh𝑌\text{\bf Sh}(Y); again this can be checked with Lemma 3.2.2.

3.3. Rings with Involution

Throughout, R𝑅R is an ordinary commutative ring, λ:RR:𝜆𝑅𝑅\lambda:R\to R is an involution, and S𝑆S is the fixed ring of λ𝜆\lambda. The purpose of this section is twofold. First, we show that the locus of primes 𝔭SpecS𝔭Spec𝑆{\mathfrak{p}}\in\operatorname{Spec}S such that R𝔭subscript𝑅𝔭R_{\mathfrak{p}} is a quadratic étale over S𝔭subscript𝑆𝔭S_{\mathfrak{p}} is open in SpecSSpec𝑆\operatorname{Spec}S. Second, we study the structure of R𝑅R when S𝑆S is local, showing, in particular, that R𝑅R is quadratic étale over S𝑆S, or R𝑅R is local.

There are two pitfalls in the study of R𝑅R over S𝑆S. First of all, R𝑅R may not be finite over S𝑆S.

Example 3.3.1.

Let I𝐼I be any set, let R𝑅R be the commutative \mathbb{C}-algebra freely generated by {xi}iIsubscriptsubscript𝑥𝑖𝑖𝐼\{x_{i}\}_{i\in I}, and let λ:RR:𝜆𝑅𝑅\lambda:R\to R be the \mathbb{C}-linear involution sending each xisubscript𝑥𝑖x_{i} to xisubscript𝑥𝑖-x_{i}. Then the fixed ring of λ𝜆\lambda is S=[xixj|i,jI]𝑆delimited-[]conditionalsubscript𝑥𝑖subscript𝑥𝑗𝑖𝑗𝐼S=\mathbb{C}[x_{i}x_{j}\,|\,i,j\in I]. Let 𝔪=xixj|i,jI𝔪inner-productsubscript𝑥𝑖subscript𝑥𝑗𝑖𝑗𝐼{\mathfrak{m}}=\langle x_{i}x_{j}\,|\,i,j\in I\rangle. Since R/𝔪R[xi|iI]/xixj|i,jI𝑅𝔪𝑅delimited-[]conditionalsubscript𝑥𝑖𝑖𝐼inner-productsubscript𝑥𝑖subscript𝑥𝑗𝑖𝑗𝐼R/{\mathfrak{m}}R\cong{\mathbb{C}[x_{i}\,|\,i\in I]}/\langle x_{i}x_{j}\,|\,i,j\in I\rangle and S/𝔪𝑆𝔪S/{\mathfrak{m}}\cong\mathbb{C}, it follows that R𝑅R cannot be generated by fewer than |I|𝐼|I| elements as an S𝑆S-algebra. Thus, when I𝐼I is infinite, R𝑅R is not finite over S𝑆S. The same applies to the S/𝔪𝑆𝔪S/{\mathfrak{m}}-algebra R/𝔪R𝑅𝔪𝑅R/{\mathfrak{m}}R, even though S/𝔪𝑆𝔪S/{\mathfrak{m}} is noetherian. We further note that when 1<|I|<01𝐼subscript01<|I|<\aleph_{0}, the ring R𝑅R is a smooth affine \mathbb{C}-algebra, but S𝑆S is singular.

Second, the formation of fixed rings may not commute with extension of scalars. That is, if Ssuperscript𝑆S^{\prime} is a commutative S𝑆S-algebra, then Ssuperscript𝑆S^{\prime} need not be the subring of λ𝜆\lambda-fixed elements in R:=RSSassignsuperscript𝑅subscripttensor-product𝑆𝑅superscript𝑆R^{\prime}:=R\otimes_{S}S^{\prime}. In fact, SRsuperscript𝑆superscript𝑅S^{\prime}\to R^{\prime} is a priori not one-to-one. Nevertheless, SRsuperscript𝑆superscript𝑅S^{\prime}\to R^{\prime} restricts to an isomorphism S{rR:r=rλ}superscript𝑆conditional-set𝑟superscript𝑅𝑟superscript𝑟𝜆S^{\prime}\to\{r\in R^{\prime}\,:\,r=r^{\lambda}\} if Ssuperscript𝑆S^{\prime} is flat over S𝑆S, or 2S×2superscript𝑆2\in{S^{\times}}. To see this, consider the exact sequence of S𝑆S-modules 0SRidSλR0𝑆𝑅subscriptid𝑆𝜆𝑅0\to S\to R\xrightarrow{\mathrm{id}_{S}-\lambda}R. The statement amounts to showing that it remains exact after tensoring with Ssuperscript𝑆S^{\prime}. This is clear if Ssuperscript𝑆S^{\prime} is flat, and if 2S×2superscript𝑆2\in{S^{\times}}, then it follows because SR𝑆𝑅S\to R is split by r12(r+rλ)maps-to𝑟12𝑟superscript𝑟𝜆r\mapsto\frac{1}{2}(r+r^{\lambda}).

Remark 3.3.2.

Voight [voight_2011_standard_involution, Corollary 3.2] showed that if R𝑅R is locally free of rank at least 333 over S𝑆S and 222 is not a zerodivisor in S𝑆S, then R𝑅R decomposes as SMdirect-sum𝑆𝑀S\oplus M where M𝑀M is an ideal of R𝑅R such that M2=0superscript𝑀20M^{2}=0 and λ|M=idMevaluated-at𝜆𝑀subscriptid𝑀\lambda|_{M}=-\mathrm{id}_{M}. Voight calls such (commutative) rings with involution exceptional. This shows that if λ:RR:𝜆𝑅𝑅\lambda:R\to R is not exceptional then either R𝑅R is not locally free over S𝑆S, or rankSR2subscriptrank𝑆𝑅2\operatorname{rank}_{S}R\leq 2. The case R=[xi|iI]/xixj|i,jI𝑅delimited-[]conditionalsubscript𝑥𝑖𝑖𝐼inner-productsubscript𝑥𝑖subscript𝑥𝑗𝑖𝑗𝐼R=\mathbb{C}[x_{i}\,|\,i\in I]/\langle x_{i}x_{j}\,|\,i,j\in I\rangle and S=𝑆S=\mathbb{C} featuring in Example 3.3.1 is an example of an exceptional ring with involution, and essentially the only one if S=𝑆S=\mathbb{C}.

Having warned the reader of these pitfalls, we return to the main topic of the section, which is the study of R𝑅R over S𝑆S.

Lemma 3.3.3.

Assume that there exists rR𝑟𝑅r\in R with rrλR×𝑟superscript𝑟𝜆superscript𝑅r-r^{\lambda}\in{R^{\times}}. Then R𝑅R is a quadratic étale S𝑆S-algebra.

Proof.

For aR𝑎𝑅a\in R, write ta=a+aλsubscript𝑡𝑎𝑎superscript𝑎𝜆t_{a}=a+a^{\lambda} and na=aλasubscript𝑛𝑎superscript𝑎𝜆𝑎n_{a}=a^{\lambda}a, and observe that ta,naSsubscript𝑡𝑎subscript𝑛𝑎𝑆t_{a},n_{a}\in S and a2taa+na=0superscript𝑎2subscript𝑡𝑎𝑎subscript𝑛𝑎0a^{2}-t_{a}a+n_{a}=0.

Suppose that R=S[r]𝑅𝑆delimited-[]𝑟R=S[r]. Since r2trr+nr=0superscript𝑟2subscript𝑡𝑟𝑟subscript𝑛𝑟0r^{2}-t_{r}r+n_{r}=0, it follows that R=S+Sr𝑅𝑆𝑆𝑟R=S+Sr. Furthermore, if αrSr𝛼𝑟𝑆𝑟\alpha r\in Sr for αS𝛼𝑆\alpha\in S, then 0=(αr)(αr)λ=α(rrλ)0𝛼𝑟superscript𝛼𝑟𝜆𝛼𝑟superscript𝑟𝜆0=(\alpha r)-(\alpha r)^{\lambda}=\alpha(r-r^{\lambda}), so α=0𝛼0\alpha=0 because rrλR×𝑟superscript𝑟𝜆superscript𝑅r-r^{\lambda}\in{R^{\times}}. It follows that the S𝑆S-algebra map RS[x]/(x2trx+nr)𝑅𝑆delimited-[]𝑥superscript𝑥2subscript𝑡𝑟𝑥subscript𝑛𝑟R\to S[x]/(x^{2}-t_{r}x+n_{r}) sending x𝑥x to r𝑟r is an isomorphism. Since tr24nr=(rrλ)2S×superscriptsubscript𝑡𝑟24subscript𝑛𝑟superscript𝑟superscript𝑟𝜆2superscript𝑆t_{r}^{2}-4n_{r}=(r-r^{\lambda})^{2}\in{S^{\times}}, we conclude that R𝑅R is a quadratic étale S𝑆S-algebra.

We now show that R=S[r]𝑅𝑆delimited-[]𝑟R=S[r]. Write u=rrλ=2rtrS[r]𝑢𝑟superscript𝑟𝜆2𝑟subscript𝑡𝑟𝑆delimited-[]𝑟u=r-r^{\lambda}=2r-t_{r}\in S[r] and a=u1r𝑎superscript𝑢1𝑟a=u^{-1}r. One verifies that a=nu1(nrr2)𝑎superscriptsubscript𝑛𝑢1subscript𝑛𝑟superscript𝑟2a=n_{u}^{-1}(n_{r}-r^{2}), so that aS[r]𝑎𝑆delimited-[]𝑟a\in S[r]. Since uλ=usuperscript𝑢𝜆𝑢u^{\lambda}=-u, we have a+aλ=u1ru1rλ=u1u=1𝑎superscript𝑎𝜆superscript𝑢1𝑟superscript𝑢1superscript𝑟𝜆superscript𝑢1𝑢1a+a^{\lambda}=u^{-1}r-u^{-1}r^{\lambda}=u^{-1}u=1. Let bR𝑏𝑅b\in R and b=ba(b+bλ)superscript𝑏𝑏𝑎𝑏superscript𝑏𝜆b^{\prime}=b-a(b+b^{\lambda}). Straightforward computation shows that bλ=bsuperscript𝑏𝜆superscript𝑏b^{\prime\lambda}=-b^{\prime}, hence (u1b)λ=u1bsuperscriptsuperscript𝑢1superscript𝑏𝜆superscript𝑢1superscript𝑏(u^{-1}b^{\prime})^{\lambda}=u^{-1}b^{\prime} and u1bSsuperscript𝑢1superscript𝑏𝑆u^{-1}b^{\prime}\in S. It follows that buSS[r]superscript𝑏𝑢𝑆𝑆delimited-[]𝑟b^{\prime}\in uS\subseteq S[r] and thus, b=b+a(b+bλ)S[r]+aS=S[r]𝑏superscript𝑏𝑎𝑏superscript𝑏𝜆𝑆delimited-[]𝑟𝑎𝑆𝑆delimited-[]𝑟b=b^{\prime}+a(b+b^{\lambda})\in S[r]+aS=S[r]. ∎

Lemma 3.3.4.

Suppose that S𝑆S is local and R𝑅R is a quadratic étale S𝑆S-algebra. Then there exists rR𝑟𝑅r\in R such that rrλR×𝑟superscript𝑟𝜆superscript𝑅r-r^{\lambda}\in{R^{\times}}.

Proof.

Let 𝔪𝔪{\mathfrak{m}} be the maximal ideal of S𝑆S. By Proposition 3.1.4(i), we may replace R𝑅R with R/𝔪R𝑅𝔪𝑅R/{\mathfrak{m}}R. The claim then follows easily from Proposition 3.1.4(ii). ∎

Corollary 3.3.5.

The set of prime ideals 𝔭SpecS𝔭Spec𝑆{\mathfrak{p}}\in\operatorname{Spec}S such that R𝔭subscript𝑅𝔭R_{\mathfrak{p}} is a quadratic étale S𝔭subscript𝑆𝔭S_{\mathfrak{p}}-algebra is open in SpecSSpec𝑆\operatorname{Spec}S. Equivalently for every 𝔭SpecS𝔭Spec𝑆{\mathfrak{p}}\in\operatorname{Spec}S such that R𝔭subscript𝑅𝔭R_{\mathfrak{p}} is a quadratic étale S𝔭subscript𝑆𝔭S_{\mathfrak{p}}-algebra, there exists sS𝔭𝑠𝑆𝔭s\in S-{\mathfrak{p}} such that Rssubscript𝑅𝑠R_{s} is a quadratic étale Sssubscript𝑆𝑠S_{s}-algebra.

Proof.

By Lemma 3.3.4, there exists rR𝔭𝑟subscript𝑅𝔭r\in{R_{\mathfrak{p}}} with rrλR𝔭×𝑟superscript𝑟𝜆superscriptsubscript𝑅𝔭r-r^{\lambda}\in{R_{\mathfrak{p}}^{\times}}. We can find s1S𝔭subscript𝑠1𝑆𝔭s_{1}\in S-{\mathfrak{p}} and a,bR𝑎𝑏𝑅a,b\in R such that r=as11𝑟𝑎superscriptsubscript𝑠11r=as_{1}^{-1} and rrλ=bs11𝑟superscript𝑟𝜆𝑏superscriptsubscript𝑠11r-r^{\lambda}=bs_{1}^{-1} in R𝔭subscript𝑅𝔭R_{\mathfrak{p}}. Since bs11R𝔭×𝑏superscriptsubscript𝑠11superscriptsubscript𝑅𝔭bs_{1}^{-1}\in{R_{\mathfrak{p}}^{\times}}, we can find s2S𝔭subscript𝑠2𝑆𝔭s_{2}\in S-{\mathfrak{p}} such that bs11𝑏superscriptsubscript𝑠11bs_{1}^{-1} is invertible in Rs1s2subscript𝑅subscript𝑠1subscript𝑠2R_{s_{1}s_{2}}. Now take s=s1s2𝑠subscript𝑠1subscript𝑠2s=s_{1}s_{2} and apply Lemma 3.3.3 to Rssubscript𝑅𝑠R_{s} with r=as11𝑟𝑎superscriptsubscript𝑠11r=as_{1}^{-1}. ∎

Remark 3.3.6.

Corollary 3.3.5 is well known when R𝑅R is locally free of finite rank over S𝑆S; see, for instance, [de_jong_stacks_2017, Tag 0C3J].

We momentarily consider an arbitrary finite group acting on R𝑅R.

Proposition 3.3.7.

Let G𝐺G be a finite group acting on a ring R𝑅R and let S𝑆S be the subring of elements fixed under G𝐺G. If S𝑆S is local then the maximal ideals of R𝑅R form a single G𝐺G-orbit. In particular, R𝑅R is semilocal.

Proof.

Let 𝔪𝔪{\mathfrak{m}} denote the maximal ideal of S𝑆S and, for the sake of contradiction, suppose 𝔭𝔭{\mathfrak{p}} and 𝔮𝔮{\mathfrak{q}} are maximal ideals of R𝑅R lying in distinct G𝐺G-orbits. Let 𝔭=gGg(𝔭)superscript𝔭subscript𝑔𝐺𝑔𝔭{\mathfrak{p}}^{\prime}=\bigcap_{g\in G}g({\mathfrak{p}}) and 𝔮=gGg(𝔮)superscript𝔮subscript𝑔𝐺𝑔𝔮{\mathfrak{q}}^{\prime}=\bigcap_{g\in G}g({\mathfrak{q}}). Since g(𝔭)+h(𝔮)=R𝑔𝔭𝔮𝑅g({\mathfrak{p}})+h({\mathfrak{q}})=R for all g,hG𝑔𝐺g,h\in G, we have 𝔭+𝔮=Rsuperscript𝔭superscript𝔮𝑅{\mathfrak{p}}^{\prime}+{\mathfrak{q}}^{\prime}=R. Using the Chinese Remainder Theorem, choose r𝔮𝑟superscript𝔮r\in{\mathfrak{q}}^{\prime} such that r1mod𝔭𝑟modulo1superscript𝔭r\equiv 1\!\!\mod{\mathfrak{p}}^{\prime}. Replacing r𝑟r with gGg(r)subscriptproduct𝑔𝐺𝑔𝑟\prod_{g\in G}g(r), we may assume that rS𝑟𝑆r\in S. Since 𝔭Ssuperscript𝔭𝑆{\mathfrak{p}}^{\prime}\cap S and 𝔮Ssuperscript𝔮𝑆{\mathfrak{q}}^{\prime}\cap S are both contained in 𝔪𝔪{\mathfrak{m}}, this means that r𝑟r lies both in (1+𝔭)S1+𝔪1superscript𝔭𝑆1𝔪(1+{\mathfrak{p}}^{\prime})\cap S\subseteq 1+{\mathfrak{m}} and in 𝔮S𝔪superscript𝔮𝑆𝔪{\mathfrak{q}}^{\prime}\cap S\subseteq{\mathfrak{m}}, which is absurd. ∎

We derive the main result of this section by specializing Proposition 3.3.7 to the case of a group with 222 elements.

Theorem 3.3.8.

Suppose R𝑅R is a ring and λ:RR:𝜆𝑅𝑅\lambda:R\to R is an involution with fixed ring S𝑆S such that S𝑆S is local. Let R¯¯𝑅\overline{R} denote R/Jac(R)𝑅Jac𝑅R/\mathrm{Jac}(R) and λ¯¯𝜆\overline{\lambda} the restriction of λ𝜆\lambda to R¯¯𝑅\overline{R}.

  1. (i)

    If λ¯id¯𝜆id\overline{\lambda}\neq\mathrm{id}, then R𝑅R is a quadratic étale algebra over S𝑆S.

  2. (ii)

    If λ¯=id¯𝜆id\overline{\lambda}=\mathrm{id}, then R𝑅R is a local ring that is not quadratic étale over S𝑆S.

In either case, R𝑅R is semilocal.

Proof.

Let 𝔐𝔐{\mathfrak{M}} be a maximal ideal of R𝑅R. Taking G={1,λ}𝐺1𝜆G=\{1,\lambda\} in Proposition 3.3.7, we see that the maximal ideals of R𝑅R are {𝔐,𝔐λ}𝔐superscript𝔐𝜆\{{\mathfrak{M}},{\mathfrak{M}}^{\lambda}\}. We consider the cases 𝔐𝔐λ𝔐superscript𝔐𝜆{\mathfrak{M}}\neq{\mathfrak{M}}^{\lambda} and 𝔐=𝔐λ𝔐superscript𝔐𝜆{\mathfrak{M}}={\mathfrak{M}}^{\lambda} separately.

Suppose that 𝔐𝔐λ𝔐superscript𝔐𝜆{\mathfrak{M}}\neq{\mathfrak{M}}^{\lambda}. By the Chinese Remainder Theorem, R¯=R/(𝔐𝔐λ)R/𝔐×R/𝔐λ¯𝑅𝑅𝔐superscript𝔐𝜆𝑅𝔐𝑅superscript𝔐𝜆\overline{R}=R/({\mathfrak{M}}\cap{\mathfrak{M}}^{\lambda})\cong R/{\mathfrak{M}}\times R/{\mathfrak{M}}^{\lambda}, and under this isomorphism, λ¯¯𝜆\overline{\lambda} acts by sending (a+𝔐,b+𝔐λ)𝑎𝔐𝑏superscript𝔐𝜆(a+{\mathfrak{M}},b+{\mathfrak{M}}^{\lambda}) to (bλ+𝔐,aλ+𝔐λ)superscript𝑏𝜆𝔐superscript𝑎𝜆superscript𝔐𝜆(b^{\lambda}+{\mathfrak{M}},a^{\lambda}+{\mathfrak{M}}^{\lambda}). This implies that λ¯id¯𝜆id\overline{\lambda}\neq\mathrm{id}, so we are in the situation of (i). Furthermore, by taking a=1𝑎1a=1 and b=0𝑏0b=0, we see that there exists r¯R¯𝑟𝑅\overline{r}\in R such that rrλ¯R¯×¯𝑟superscript𝑟𝜆superscript¯𝑅\overline{r-r^{\lambda}}\in{\overline{R}^{\times}}, or equivalently, rrλR×𝑟superscript𝑟𝜆superscript𝑅r-r^{\lambda}\in{R^{\times}}. Thus, by Lemma 3.3.3, R𝑅R is quadratic étale over S𝑆S.

Suppose now that 𝔐=𝔐λ𝔐superscript𝔐𝜆{\mathfrak{M}}={\mathfrak{M}}^{\lambda}. Then R𝑅R is local and R¯=R/𝔐¯𝑅𝑅𝔐\overline{R}=R/{\mathfrak{M}} is a field. If λ¯id¯𝜆id\overline{\lambda}\neq\mathrm{id}, then there exists r¯R¯¯𝑟¯𝑅\overline{r}\in\overline{R} with rrλ¯R¯×¯𝑟superscript𝑟𝜆superscript¯𝑅\overline{r-r^{\lambda}}\in{\overline{R}^{\times}} and again we find that R𝑅R is quadratic étale over S𝑆S. On the other hand, if λ¯=id¯𝜆id\overline{\lambda}=\mathrm{id}, then R𝑅R cannot be quadratic étale over S𝑆S by Proposition 3.1.4(ii).

We have verified (i) and (ii) in both cases, so the proof is complete. ∎

Notation 3.3.9.

A henselian ring is a local ring in which Hensel’s lemma, [eisenbud_commutative_1995, Thm. 7.3], holds. A strictly henselian ring is a henselian ring for which the residue field is separably closed.

Lemma 3.3.10.

Let G𝐺G be a finite group acting on a ring R𝑅R and let S𝑆S be the subring of R𝑅R fixed under G𝐺G. If S𝑆S is local with maximal ideal 𝔪𝔪{\mathfrak{m}}, then 𝔪RJac(R)𝔪𝑅Jac𝑅{\mathfrak{m}}R\subseteq\mathrm{Jac}(R). In particular, 𝔪RS=𝔪𝔪𝑅𝑆𝔪{\mathfrak{m}}R\cap S={\mathfrak{m}}.

Proof.

Let a𝔪𝑎𝔪a\in{\mathfrak{m}}. To prove 𝔪RJac(R)𝔪𝑅Jac𝑅{\mathfrak{m}}R\subseteq\mathrm{Jac}(R), we need to show that aRJac(R)𝑎𝑅Jac𝑅aR\subseteq\mathrm{Jac}(R), or equivalently, that 1+aR1𝑎𝑅1+aR consists of invertible elements. Let bR𝑏𝑅b\in R and let {g1,,gn}subscript𝑔1subscript𝑔𝑛\{g_{1},\dots,g_{n}\} denote the distinct elements of G𝐺G. Then

gGg(1+ab)=1+σ1(g1b,,gnb)a+σ2(g1b,,gnb)a2+subscriptproduct𝑔𝐺𝑔1𝑎𝑏1subscript𝜎1subscript𝑔1𝑏subscript𝑔𝑛𝑏𝑎subscript𝜎2subscript𝑔1𝑏subscript𝑔𝑛𝑏superscript𝑎2\prod_{g\in G}g(1+ab)=1+\sigma_{1}(g_{1}b,\dots,g_{n}b)a+\sigma_{2}(g_{1}b,\dots,g_{n}b)a^{2}+\dots

where σisubscript𝜎𝑖\sigma_{i} denotes the i𝑖i-th elementary symmetric polynomial on n𝑛n letters. Since a𝔪𝑎𝔪a\in{\mathfrak{m}}, and since σi(g1b,,gnb)subscript𝜎𝑖subscript𝑔1𝑏subscript𝑔𝑛𝑏\sigma_{i}(g_{1}b,\dots,g_{n}b) is invariant under G𝐺G, the right hand side lies in 1+𝔪S×1𝔪superscript𝑆1+{\mathfrak{m}}\subseteq{S^{\times}}. Thus, 1+abR×1𝑎𝑏superscript𝑅1+ab\in{R^{\times}}. ∎

Corollary 3.3.11.

Let R𝑅R be a ring, let λ:RR:𝜆𝑅𝑅\lambda:R\to R be an involution, and let S𝑆S denote the fixed ring of λ𝜆\lambda. Suppose that S𝑆S is a strictly henselian ring with maximal ideal 𝔪𝔪{\mathfrak{m}}. Then R𝑅R is a finite product of strictly henselian rings.

Proof.

By Theorem 3.3.8, either R𝑅R is a quadratic étale S𝑆S-algebra, or R𝑅R is local. In the former case, R𝑅R is finite over S𝑆S, and the lemma follows from [de_jong_stacks_2017, Tag 04GH] so we assume R𝑅R is local. Write 𝔐𝔐{\mathfrak{M}} for the maximal ideal of R𝑅R, k𝑘k for its residue field, and denote by rr¯maps-to𝑟¯𝑟r\mapsto\overline{r} the surjection Rk𝑅𝑘R\to k.

Observe first that R𝑅R is integral over S𝑆S since for all rR𝑟𝑅r\in R, we have r2(rλ+r)r+rλr=0superscript𝑟2superscript𝑟𝜆𝑟𝑟superscript𝑟𝜆𝑟0r^{2}-(r^{\lambda}+r)r+r^{\lambda}r=0 and rλ+r,rλrSsuperscript𝑟𝜆𝑟superscript𝑟𝜆𝑟𝑆r^{\lambda}+r,r^{\lambda}r\in S. This implies that k𝑘k is algebraic over the residue field of S𝑆S, which is separably closed, hence k𝑘k is separably closed.

Now, let fR[x]𝑓𝑅delimited-[]𝑥f\in R[x] be a monic polynomial such that f¯k[x]¯𝑓𝑘delimited-[]𝑥\overline{f}\in k[x] has a simple root ηk𝜂𝑘\eta\in k. We will show that f𝑓f has root yR𝑦𝑅y\in R with y¯=η¯𝑦𝜂\overline{y}=\eta. Let a0,,an1Rsubscript𝑎0subscript𝑎𝑛1𝑅a_{0},\dots,a_{n-1}\in R be the coefficients of f𝑓f and let xR𝑥𝑅x\in R be any element with x¯=η¯𝑥𝜂\overline{x}=\eta. Since R𝑅R is integral over S𝑆S, there is a finite S𝑆S-subalgebra R0Rsubscript𝑅0𝑅R_{0}\subseteq R containing a0,,an1,xsubscript𝑎0subscript𝑎𝑛1𝑥a_{0},\dots,a_{n-1},x. By [de_jong_stacks_2017, Tag 04GH], R0subscript𝑅0R_{0} is a product of henselian rings. Since R𝑅R has no non-trivial idempotents, this means R0subscript𝑅0R_{0} is a henselian ring. Write 𝔐0subscript𝔐0{\mathfrak{M}}_{0} for the maximal ideal of R0subscript𝑅0R_{0} and k0=R0/𝔐0subscript𝑘0subscript𝑅0subscript𝔐0k_{0}=R_{0}/{\mathfrak{M}}_{0}. Since R0subscript𝑅0R_{0} is finite over S𝑆S, there is a natural number n𝑛n such that 𝔐0n𝔪R0𝔐0superscriptsubscript𝔐0𝑛𝔪subscript𝑅0subscript𝔐0{\mathfrak{M}}_{0}^{n}\subseteq{\mathfrak{m}}R_{0}\subseteq{\mathfrak{M}}_{0} by [reiner_maximal_1975-1, Th. 6.15]. This implies that (R𝔐0)n=R𝔐0nR𝔪superscript𝑅subscript𝔐0𝑛𝑅superscriptsubscript𝔐0𝑛𝑅𝔪(R{\mathfrak{M}}_{0})^{n}=R{\mathfrak{M}}_{0}^{n}\subseteq R{\mathfrak{m}} and the latter is a proper ideal of R𝑅R, by Lemma 3.3.10. Therefore, R𝔐0𝔐𝑅subscript𝔐0𝔐R{\mathfrak{M}}_{0}\subseteq{\mathfrak{M}} and we have a well-defined homomorphism of fields k0ksubscript𝑘0𝑘k_{0}\to k given by x+𝔐0x+𝔐maps-to𝑥subscript𝔐0𝑥𝔐x+{\mathfrak{M}}_{0}\mapsto x+{\mathfrak{M}}. Since a0,,an1,xR0subscript𝑎0subscript𝑎𝑛1𝑥subscript𝑅0a_{0},\dots,a_{n-1},x\in R_{0}, we have f¯k0[x]¯𝑓subscript𝑘0delimited-[]𝑥\overline{f}\in k_{0}[x] and ηk0𝜂subscript𝑘0\eta\in k_{0}. As R0subscript𝑅0R_{0} is a henselian ring, there is yR0𝑦subscript𝑅0y\in R_{0} with f(y)=0𝑓𝑦0f(y)=0 and y+𝔐0=η𝑦subscript𝔐0𝜂y+{\mathfrak{M}}_{0}=\eta. This completes the proof. ∎

4. Ringed Topoi with Involution

Unless indicated otherwise, throughout this section, (𝐗,𝒪𝐗)𝐗subscript𝒪𝐗(\mathbf{X},{\mathcal{O}}_{\mathbf{X}}) denotes a locally ringed topos. Our interest is in the following examples:

  1. (1)

    𝐗=Sh(Xét)𝐗Shsubscript𝑋ét\mathbf{X}=\text{\bf Sh}(X_{\text{\'{e}t}}) for a scheme X𝑋X, and 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}} is the structure sheaf of X𝑋X which sends (UX)𝑈𝑋(U\to X) to Γ(U,𝒪U)Γ𝑈subscript𝒪𝑈\Gamma(U,{\mathcal{O}}_{U}).

  2. (2)

    𝐗=Sh(X)𝐗Sh𝑋\mathbf{X}=\text{\bf Sh}(X) for a topological space X𝑋X, and 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}} is the sheaf of continuous \mathbb{C}-valued functions, denoted 𝒞(X,)𝒞𝑋{\mathcal{C}}(X,\mathbb{C}).

We write the cyclic group with two elements as C2subscript𝐶2C_{2} and, when applicable, the non-trivial element of C2subscript𝐶2C_{2} will always be denoted λ𝜆\lambda.

When there is no risk of confusion, we shall refer to 𝐗𝐗{\mathbf{X}} as a ringed topos, in which case the ring object is understood to be 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}}.

4.1. Involutions of Ringed Topoi

Definition 4.1.1.

Suppose 𝐗𝐗\mathbf{X} is a topos. An involution of 𝐗𝐗\mathbf{X} consists of an equivalence of categories Λ:𝐗𝐗:Λ𝐗𝐗\Lambda:\mathbf{X}\to\mathbf{X} and a natural isomorphism ν:Λ2id:𝜈superscriptΛ2id\nu:\Lambda^{2}\Rightarrow\mathrm{id} satisfying the coherence condition that νΛX=ΛνXsubscript𝜈Λ𝑋Λsubscript𝜈𝑋\nu_{\Lambda X}=\Lambda\nu_{X} for all objects X𝑋X of 𝐗𝐗\mathbf{X}.

The natural equivalence ν𝜈\nu will generally be suppressed from the notation.

Definition 4.1.2.

Suppose (𝐗,𝒪𝐗)𝐗subscript𝒪𝐗(\mathbf{X},\mathcal{O}_{\mathbf{X}}) is a ringed topos. An involution of (𝐗,𝒪𝐗)𝐗subscript𝒪𝐗(\mathbf{X},{\mathcal{O}}_{\mathbf{X}}) consists of an involution (Λ,ν)Λ𝜈(\Lambda,\nu) of 𝐗𝐗\mathbf{X} and an isomorphism λ𝜆\lambda of ring objects λ:𝒪𝐗Λ𝒪𝐗:𝜆subscript𝒪𝐗Λsubscript𝒪𝐗\lambda:{\mathcal{O}}_{\mathbf{X}}\to\Lambda{\mathcal{O}}_{\mathbf{X}} such that Λλλ=ν𝒪𝐗1Λ𝜆𝜆superscriptsubscript𝜈subscript𝒪𝐗1\Lambda\lambda\circ\lambda=\nu_{{\mathcal{O}}_{\mathbf{X}}}^{-1}.

Suppressing ν𝜈\nu from the last equation, we say Λλλ=idΛ𝜆𝜆id\Lambda\lambda\circ\lambda=\mathrm{id}.

Remark 4.1.3.

(i) The functor ΛΛ\Lambda is left adjoint to itself with the unit and counit of the adjunction being ν1superscript𝜈1\nu^{-1} and ν𝜈\nu, respectively. Thus, if we write Λ=Λ=ΛsuperscriptΛsubscriptΛΛ\Lambda^{*}=\Lambda_{*}=\Lambda, then the adjoint pair (Λ,Λ)superscriptΛsubscriptΛ(\Lambda^{*},\Lambda_{*}) defines a geometric automorphism of 𝐗𝐗\mathbf{X}. Moreover, (Λ,Λ,λ1:Λ𝒪𝐗𝒪𝐗):superscriptΛsubscriptΛsuperscript𝜆1subscriptΛsubscript𝒪𝐗subscript𝒪𝐗(\Lambda^{*},\Lambda_{*},\lambda^{-1}:\Lambda_{*}{\mathcal{O}}_{\mathbf{X}}\to{\mathcal{O}}_{\mathbf{X}}) defines an automorphism of the ringed topos (𝐗,𝒪𝐗)𝐗subscript𝒪𝐗({\mathbf{X}},{\mathcal{O}}_{\mathbf{X}}).

(ii) Topoi and ringed topoi form 222-categories in which there is a notion of a weak C2subscript𝐶2C_{2}-action. An involution of a topos, resp. ringed topos, as defined here induces such a weak C2subscript𝐶2C_{2}-action in which the non-trivial element λ𝜆\lambda acts as the morphism Λ=(Λ,Λ)ΛsuperscriptΛsubscriptΛ\Lambda=(\Lambda^{*},\Lambda_{*}), resp. (Λ,Λ,λ1)superscriptΛsubscriptΛsuperscript𝜆1(\Lambda^{*},\Lambda_{*},\lambda^{-1}), and the trivial element acts as the identity. It can be checked that all C2subscript𝐶2C_{2}-actions with the latter property arise in this manner. Since an arbitrary weak C2subscript𝐶2C_{2}-action is equivalent to one in which 111 acts as the identity, specifying an involution is essentially the same as specifying a weak C2subscript𝐶2C_{2}-action.

Notation 4.1.4.

Henceforth, when there is no risk of confusion, involutions of ringed topoi will always be denoted (Λ,ν,λ)Λ𝜈𝜆(\Lambda,\nu,\lambda). In fact, we shall often abbreviate the triple (Λ,ν,λ)Λ𝜈𝜆(\Lambda,\nu,\lambda) to λ𝜆\lambda.

It is convenient to think of Λ,νΛ𝜈\Lambda,\nu as the “geometric data” of the involution and of λ𝜆\lambda as the “arithmetic data” of the involution. The following are the motivating examples.

Example 4.1.5.

Let X𝑋X be a scheme and let σ:XX:𝜎𝑋𝑋\sigma:X\to X be an involution. The direct image functor Λ:=σassignΛsubscript𝜎\Lambda:=\sigma_{*} defines an involution of 𝐗:=Sh(Xét)assign𝐗Shsubscript𝑋ét{\mathbf{X}}:=\text{\bf Sh}(X_{\text{\'{e}t}}); the suppressed natural isomorphism ν:Λ2id:𝜈superscriptΛ2id\nu:\Lambda^{2}\Rightarrow\mathrm{id} is the identity. Let 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}} be the structure sheaf of X𝑋X on Xétsubscript𝑋étX_{\text{\'{e}t}}. The involution of X𝑋X gives rise to an isomorphism λ:𝒪𝐗Λ𝒪𝐗:𝜆subscript𝒪𝐗Λsubscript𝒪𝐗\lambda:{\mathcal{O}}_{\mathbf{X}}\to\Lambda{\mathcal{O}}_{\mathbf{X}} as follows: For an étale morphism UX𝑈𝑋U\to X, define UσXsuperscript𝑈𝜎𝑋U^{\sigma}\to X via the pullback diagram

Uσsuperscript𝑈𝜎\textstyle{U^{\sigma}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}σUsubscript𝜎𝑈\scriptstyle{\sigma_{U}}X𝑋\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}σ𝜎\scriptstyle{\sigma}U𝑈\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces}X𝑋\textstyle{X}

By definition, Λ𝒪𝐗(UX)=σ𝒪𝐗(UX)=𝒪𝐗(UσX)=Γ(Uσ,𝒪Uσ)Λsubscript𝒪𝐗𝑈𝑋subscript𝜎subscript𝒪𝐗𝑈𝑋subscript𝒪𝐗superscript𝑈𝜎𝑋Γsuperscript𝑈𝜎subscript𝒪superscript𝑈𝜎\Lambda{\mathcal{O}}_{\mathbf{X}}(U\to X)=\sigma_{*}{\mathcal{O}}_{\mathbf{X}}(U\to X)={\mathcal{O}}_{\mathbf{X}}(U^{\sigma}\to X)=\Gamma(U^{\sigma},{\mathcal{O}}_{U^{\sigma}}), and we define λUX:Γ(U,𝒪U)=𝒪𝐗(UX)Λ𝒪𝐗(UX)=Γ(Uσ,𝒪Uσ):subscript𝜆𝑈𝑋Γ𝑈subscript𝒪𝑈subscript𝒪𝐗𝑈𝑋Λsubscript𝒪𝐗𝑈𝑋Γsuperscript𝑈𝜎subscript𝒪superscript𝑈𝜎\lambda_{U\to X}:\Gamma(U,{\mathcal{O}}_{U})={\mathcal{O}}_{\mathbf{X}}(U\to X)\to\Lambda{\mathcal{O}}_{\mathbf{X}}(U\to X)=\Gamma(U^{\sigma},{\mathcal{O}}_{U^{\sigma}}) to be the isomorphism induced by σUsubscript𝜎𝑈\sigma_{U}. It is easy to check that Λλλ=idΛ𝜆𝜆id\Lambda\lambda\circ\lambda=\mathrm{id} and so (Λ,ν,λ)Λ𝜈𝜆(\Lambda,\nu,\lambda) is an involution of (𝐗,𝒪𝐗)𝐗subscript𝒪𝐗({\mathbf{X}},{\mathcal{O}}_{\mathbf{X}}). When X=SpecR𝑋Spec𝑅X=\operatorname{Spec}R for a ring R𝑅R and σ𝜎\sigma is induced by an involution of R𝑅R, we can recover that involution from λ:𝒪𝐗Λ𝒪𝐗:𝜆subscript𝒪𝐗Λsubscript𝒪𝐗\lambda:{\mathcal{O}}_{\mathbf{X}}\to\Lambda{\mathcal{O}}_{\mathbf{X}} by taking global sections.

The small étale site Xétsubscript𝑋étX_{\text{\'{e}t}} can be replaced by other sites, for instance the site Xfppfsubscript𝑋fppfX_{\mathrm{fppf}}.

Example 4.1.6.

Let X𝑋X be topological space with a continuous involution σ:XX:𝜎𝑋𝑋\sigma:X\to X. Write 𝐗=Sh(X)𝐗Sh𝑋\mathbf{X}=\text{\bf Sh}(X) and let 𝒪𝐗=𝒞(X,)subscript𝒪𝐗𝒞𝑋{\mathcal{O}}_{\mathbf{X}}={\mathcal{C}}(X,\mathbb{C}), the ring sheaf of continuous functions into \mathbb{C}. Then the direct image functor Λ:=σassignΛsubscript𝜎\Lambda:=\sigma_{*} defines an involution of 𝐗𝐗{\mathbf{X}}; the isomorphism ν:Λ2id:𝜈superscriptΛ2id\nu:\Lambda^{2}\Rightarrow\mathrm{id} is the identity.

In particular, (Λ𝒪𝐗)(U)=𝒪𝐗(σU)=𝒞(σU,)Λsubscript𝒪𝐗𝑈subscript𝒪𝐗𝜎𝑈𝒞𝜎𝑈(\Lambda{\mathcal{O}}_{\mathbf{X}})(U)={\mathcal{O}}_{\mathbf{X}}(\sigma U)={\mathcal{C}}(\sigma U,\mathbb{C}). Let λ:𝒪𝐗Λ𝒪𝐗:𝜆subscript𝒪𝐗Λsubscript𝒪𝐗\lambda:{\mathcal{O}}_{\mathbf{X}}\to\Lambda{\mathcal{O}}_{\mathbf{X}} be given sectionwise by precomposition with σ𝜎\sigma, namely

λU:C(U,):subscript𝜆𝑈𝐶𝑈\displaystyle\lambda_{U}:{C}(U,\mathbb{C}) C(σU,),absent𝐶𝜎𝑈\displaystyle\to{C}(\sigma U,\mathbb{C}),
ϕitalic-ϕ\displaystyle\phi ϕσ.maps-toabsentitalic-ϕ𝜎\displaystyle\mapsto\phi\circ\sigma.

Then (Λ,ν,λ)Λ𝜈𝜆(\Lambda,\nu,\lambda) is an involution of (𝐗,𝒪𝐗)𝐗subscript𝒪𝐗(\mathbf{X},{\mathcal{O}}_{\mathbf{X}}).

Example 4.1.7.

Let 𝐗𝐗{\mathbf{X}} be any topos, let R𝑅R be a ring object in 𝐗𝐗{\mathbf{X}} and let λ:RR:𝜆𝑅𝑅\lambda:R\to R be an involution. Then (id𝐗,ν:=id,λ)formulae-sequenceassignsubscriptid𝐗𝜈id𝜆(\mathrm{id}_{{\mathbf{X}}},\nu:=\mathrm{id},\lambda) is an involution of (𝐗,R)𝐗𝑅({\mathbf{X}},R).

Definition 4.1.8.

The trivial involution of (𝐗,𝒪𝐗)𝐗subscript𝒪𝐗(\mathbf{X},{\mathcal{O}}_{\mathbf{X}}) is (Λ,ν,λ)=(id𝐗,id,id𝒪𝐗)Λ𝜈𝜆subscriptid𝐗idsubscriptidsubscript𝒪𝐗(\Lambda,\nu,\lambda)=(\mathrm{id}_{\mathbf{X}},\mathrm{id},\mathrm{id}_{{\mathcal{O}}_{\mathbf{X}}}). Any other involution (Λ,ν,λ)Λ𝜈𝜆(\Lambda,\nu,\lambda) of (𝐗,𝒪𝐗)𝐗subscript𝒪𝐗(\mathbf{X},{\mathcal{O}}_{\mathbf{X}}) will be said to be weakly trivial if it is equivalent to the trivial involution of (𝐗,𝒪𝐗)𝐗subscript𝒪𝐗({\mathbf{X}},{\mathcal{O}}_{\mathbf{X}}) in the following sense: There exists a natural isomorphism θ:Λid:𝜃Λid\theta:\Lambda\Rightarrow\mathrm{id} such that θ𝒪X=λ1subscript𝜃subscript𝒪𝑋superscript𝜆1\theta_{{\mathcal{O}}_{X}}=\lambda^{-1} and θXθΛX=νXsubscript𝜃𝑋subscript𝜃Λ𝑋subscript𝜈𝑋\theta_{X}\circ\theta_{\Lambda X}=\nu_{X} for all objects X𝑋X in 𝐗𝐗{\mathbf{X}}.

Example 4.1.9.

The involution of Example 4.1.5 (resp. Example 4.1.6) is trivial if and only if σ:XX:𝜎𝑋𝑋\sigma:X\to X is the identity.

Let (Λ,ν,λ)Λ𝜈𝜆(\Lambda,\nu,\lambda) be an involution of (𝐗,𝒪𝐗)𝐗subscript𝒪𝐗(\mathbf{X},{\mathcal{O}}_{\mathbf{X}}) and let M𝑀M be an 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}}-module. Then ΛMΛ𝑀\Lambda M carries a Λ𝒪𝐗Λsubscript𝒪𝐗\Lambda{\mathcal{O}}_{\mathbf{X}}-module structure. We shall always regard ΛMΛ𝑀\Lambda M as an 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}}-module by using the morphism λ:𝒪𝐗Λ𝒪𝐗:𝜆subscript𝒪𝐗Λsubscript𝒪𝐗\lambda:{\mathcal{O}}_{\mathbf{X}}\to\Lambda{\mathcal{O}}_{\mathbf{X}}, that is, by twisting the structure.

Using the equality Λλλ=ν𝒪𝐗1Λ𝜆𝜆superscriptsubscript𝜈subscript𝒪𝐗1\Lambda\lambda\circ\lambda=\nu_{{\mathcal{O}}_{\mathbf{X}}}^{-1}, one easily checks that νM:ΛΛMM:subscript𝜈𝑀ΛΛ𝑀𝑀\nu_{M}:\Lambda\Lambda M\to M is an isomorphism of 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}}-modules, which we suppress from the notation henceforth.

Notice that contrary to the case of λ𝜆\lambda-twisting of modules over ordinary rings, it is not in general true that M𝑀M can be identified with its λ𝜆\lambda-twist as an abelian group object in 𝐗𝐗\mathbf{X}. For instance, in the context of Example 4.1.5, suppose that X=SpecR𝑋Spec𝑅X=\operatorname{Spec}R and σ𝜎\sigma exchanges two maximal ideals 𝔪1,𝔪2Rsubscript𝔪1subgroup-ofsubscript𝔪2𝑅{\mathfrak{m}}_{1},{\mathfrak{m}}_{2}\lhd R, and let M1subscript𝑀1M_{1}, M2subscript𝑀2M_{2} denote the quasicoherent 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}}-modules corresponding to the R𝑅R-modules R/𝔪1𝑅subscript𝔪1R/{\mathfrak{m}}_{1} and R/𝔪2𝑅subscript𝔪2R/{\mathfrak{m}}_{2}, respectively. Then M2=ΛM1subscript𝑀2Λsubscript𝑀1M_{2}=\Lambda M_{1}, but M1subscript𝑀1M_{1} is not isomorphic to M2subscript𝑀2M_{2} as abelian group objects in 𝐗𝐗{\mathbf{X}} because M1subscript𝑀1M_{1} is supported at {𝔪1}subscript𝔪1\{{\mathfrak{m}}_{1}\} while M2subscript𝑀2M_{2} is supported at {𝔪2}subscript𝔪2\{{\mathfrak{m}}_{2}\}.

If A𝐴A is an 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}}-algebra, then ΛAΛ𝐴\Lambda A, besides being an 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}}-module, carries an 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}}-algebra structure. Letting Aopsuperscript𝐴opA^{\text{op}} denote the opposite algebra of A𝐴A, we have Λ(ΛAop)op=AΛsuperscriptΛsuperscript𝐴opop𝐴\Lambda(\Lambda A^{\text{op}})^{\text{op}}=A up to the suppressed natural isomorphism νAsubscript𝜈𝐴\nu_{A}.

Definition 4.1.10.

A λ𝜆\lambda-involution on an 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}}-algebra A𝐴A is a morphism of 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}}-algebras τ:AΛAop:𝜏𝐴Λsuperscript𝐴op\tau:A\to\Lambda A^{\text{op}} such that Λττ=idAΛ𝜏𝜏subscriptid𝐴\Lambda\tau\circ\tau=\mathrm{id}_{A}. In this case, (A,τ)𝐴𝜏(A,\tau) is called an 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}}-algebra with a λ𝜆\lambda-involution. If A𝐴A is an Azumaya algebra, it will be called an Azumaya algebra with λ𝜆\lambda-involution.

If (A,τ)𝐴𝜏(A,\tau) and (A,τ)superscript𝐴superscript𝜏(A^{\prime},\tau^{\prime}) are 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}}-algebras with λ𝜆\lambda-involutions, then a morphism from (A,τ)𝐴𝜏(A,\tau) to (A,τ)superscript𝐴superscript𝜏(A^{\prime},\tau^{\prime}) is a morphism of 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}}-algebras ϕ:AA:italic-ϕ𝐴superscript𝐴\phi:A\to A^{\prime} such that τϕ=Λϕτsuperscript𝜏italic-ϕΛitalic-ϕ𝜏\tau^{\prime}\circ\phi=\Lambda\phi\circ\tau.

Notice that applying ΛΛ\Lambda to both sides of Λττ=idAΛ𝜏𝜏subscriptid𝐴\Lambda\tau\circ\tau=\mathrm{id}_{A} gives τΛτ=idΛA𝜏Λ𝜏subscriptidΛ𝐴\tau\circ\Lambda\tau=\mathrm{id}_{\Lambda A}.

Notation 4.1.11.

The category where the objects are degree-n𝑛n Azumaya 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}}-algebras with λ𝜆\lambda-involutions and where the morphisms are isomorphisms of 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}}-algebras with λ𝜆\lambda-involutions shall be denoted Azn(𝐗,𝒪𝐗,λ)subscriptAz𝑛𝐗subscript𝒪𝐗𝜆{\text{\bf Az}}_{n}({\mathbf{X}},{\mathcal{O}}_{\mathbf{X}},\lambda), or just Azn(𝒪𝐗,λ)subscriptAz𝑛subscript𝒪𝐗𝜆{\text{\bf Az}}_{n}({\mathcal{O}}_{\mathbf{X}},\lambda).

Given a ring with involution (R,λ)𝑅𝜆(R,\lambda) and an R𝑅R-algebra A𝐴A, it is reasonable to define a λ𝜆\lambda-involution of A𝐴A to be an involution τ:AA:𝜏𝐴𝐴\tau:A\to A satisfying (ra)τ=rλaτsuperscript𝑟𝑎𝜏superscript𝑟𝜆superscript𝑎𝜏(ra)^{\tau}=r^{\lambda}a^{\tau} for all rR𝑟𝑅r\in R, aA𝑎𝐴a\in A. Following Gille [gille_gersten_2009, §1], this can be generalized to the context of schemes: Given a scheme X𝑋X, an involution σ:XX:𝜎𝑋𝑋\sigma:X\to X, and an 𝒪Xsubscript𝒪𝑋{\mathcal{O}}_{X}-algebra ASh(XZar)𝐴Shsubscript𝑋ZarA\in\text{\bf Sh}(X_{\operatorname{Zar}}), then a σ𝜎\sigma-involution of A𝐴A is an 𝒪Xsubscript𝒪𝑋{\mathcal{O}}_{X}-algebra morphism τ:AσAop:𝜏𝐴subscript𝜎superscript𝐴op\tau:A\to\sigma_{*}A^{\text{op}} such that σττ=idAsubscript𝜎𝜏𝜏subscriptid𝐴\sigma_{*}\tau\circ\tau=\mathrm{id}_{A}. We reconcile these elementary definitions with Definition 4.1.10 in the following example.

Example 4.1.12.

Let (R,λ)𝑅𝜆(R,\lambda) be a ring with involution, let X=SpecR𝑋Spec𝑅X=\operatorname{Spec}R and write σ:XX:𝜎𝑋𝑋\sigma:X\to X for the involution induced by λ𝜆\lambda. Abusing the notation, let (Λ,ν,λ)Λ𝜈𝜆(\Lambda,\nu,\lambda) denote the involution induced by σ𝜎\sigma on the étale ringed topos of X𝑋X, see Example 4.1.5.

Every R𝑅R-module M𝑀M gives rise to an 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}}-module, also denoted M𝑀M, the sections of which are given by M(SpecRSpecR)=MRR𝑀Specsuperscript𝑅Spec𝑅subscripttensor-product𝑅𝑀superscript𝑅M(\operatorname{Spec}R^{\prime}\to\operatorname{Spec}R)=M\otimes_{R}R^{\prime} for any (SpecRSpecR)Specsuperscript𝑅Spec𝑅(\operatorname{Spec}R^{\prime}\to\operatorname{Spec}R) in Xétsubscript𝑋étX_{\text{\'{e}t}}. In fact, this defines an equivalence of categories between R𝑅R-modules and quasicoherent 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}}-modules [de_jong_stacks_2017, Tag 03DX]. Straightforward computation shows that the 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}}-module ΛMΛ𝑀\Lambda M corresponds to Mλsuperscript𝑀𝜆M^{\lambda}, the R𝑅R-module obtained from M𝑀M by twisting via λ𝜆\lambda.

Now let A𝐴A be an R𝑅R-algebra and let τ:AA:𝜏𝐴𝐴\tau:A\to A be an involution satisfying (ra)τ=rλaτsuperscript𝑟𝑎𝜏superscript𝑟𝜆superscript𝑎𝜏(ra)^{\tau}=r^{\lambda}a^{\tau} for all rR𝑟𝑅r\in R, aA𝑎𝐴a\in A. Realizing A𝐴A as an 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}}-algebra in 𝐗𝐗{\mathbf{X}}, the algebra ΛAopΛsuperscript𝐴op\Lambda A^{\text{op}} corresponds to the R𝑅R-algebra (Aλ)opsuperscriptsuperscript𝐴𝜆op(A^{\lambda})^{\text{op}}, and so the involution τ𝜏\tau induces a λ𝜆\lambda-involution AΛAop𝐴Λsuperscript𝐴opA\to\Lambda A^{\text{op}}, also denoted τ𝜏\tau. By taking global sections, we see that all λ𝜆\lambda-involutions of the 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}}-algebra A𝐴A are obtained in this manner.

Likewise, if X𝑋X is a scheme, σ:XX:𝜎𝑋𝑋\sigma:X\to X is an involution, and A𝐴A is a quasicoherent 𝒪Xsubscript𝒪𝑋{\mathcal{O}}_{X}-algebra in Sh(XZar)Shsubscript𝑋Zar\text{\bf Sh}(X_{\operatorname{Zar}}), then the λ𝜆\lambda-involutions of the 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}}-algebra associated to A𝐴A in 𝐗=Sh(Xét)𝐗Shsubscript𝑋ét{\mathbf{X}}=\text{\bf Sh}(X_{\text{\'{e}t}}) (see [de_jong_stacks_2017, Tag 03DU]) are in one-to-one correspondence with the σ𝜎\sigma-involutions of A𝐴A in the sense of Gille [gille_gersten_2009, §1]. The quasicoherence assumption applies in particular when A𝐴A is an Azumaya algebra over X𝑋X.

Example 4.1.13.

Let λ=(Λ,ν,λ)𝜆Λ𝜈𝜆\lambda=(\Lambda,\nu,\lambda) be an involution of (𝐗,𝒪𝐗)𝐗subscript𝒪𝐗({\mathbf{X}},{\mathcal{O}}_{\mathbf{X}}) and let n𝑛n be a natural number. Then Mn×n(𝒪𝐗)subscriptM𝑛𝑛subscript𝒪𝐗\mathrm{M}_{n\times n}({\mathcal{O}}_{\mathbf{X}}) admits a λ𝜆\lambda-involution given by (αij)i,j(αjiλ)i,jmaps-tosubscriptsubscript𝛼𝑖𝑗𝑖𝑗subscriptsuperscriptsubscript𝛼𝑗𝑖𝜆𝑖𝑗(\alpha_{ij})_{i,j}\mapsto(\alpha_{ji}^{\lambda})_{i,j} on sections and denoted λtr𝜆tr\lambda\text{\rm tr}. If the sections αijsubscript𝛼𝑖𝑗\alpha_{ij} lie in 𝒪𝐗(U)subscript𝒪𝐗𝑈{\mathcal{O}}_{\mathbf{X}}(U), then the sections αjiλsuperscriptsubscript𝛼𝑗𝑖𝜆\alpha_{ji}^{\lambda} lie in (Λ𝒪𝐗)(U)Λsubscript𝒪𝐗𝑈(\Lambda{\mathcal{O}}_{\mathbf{X}})(U).

4.2. Morphisms

We now give the general definition of a morphism of ringed topoi with involution. Only very few examples of these will be considered in the sequel.

Recall that a geometric morphism of topoi f:𝐗𝐗:𝑓𝐗superscript𝐗f:{\mathbf{X}}\to{\mathbf{X}}^{\prime} consists of two functors f:𝐗𝐗:subscript𝑓𝐗superscript𝐗f_{*}:{\mathbf{X}}\to{\mathbf{X}}^{\prime}, f:𝐗𝐗:superscript𝑓superscript𝐗𝐗f^{*}:{\mathbf{X}}^{\prime}\to{\mathbf{X}} together with an adjunction between fsuperscript𝑓f^{*} and fsubscript𝑓f_{*} and such that fsuperscript𝑓f^{*} commutes with finite limits. We shall usually denote the unit and counit natural transformations associated to the adjunction by η(f):id𝐗ff:superscript𝜂𝑓subscriptidsuperscript𝐗subscript𝑓superscript𝑓\eta^{(f)}:\mathrm{id}_{{\mathbf{X}}^{\prime}}\Rightarrow f_{*}f^{*} and ε(f):ffid𝐗:superscript𝜀𝑓superscript𝑓subscript𝑓subscriptid𝐗\varepsilon^{(f)}:f^{*}f_{*}\Rightarrow\mathrm{id}_{{\mathbf{X}}}, dropping the superscript f𝑓f when there is no risk of confusion. If (𝐗,𝒪)𝐗𝒪({\mathbf{X}},{\mathcal{O}}) and (𝐗,𝒪)superscript𝐗superscript𝒪({\mathbf{X}}^{\prime},{\mathcal{O}}^{\prime}) are ringed topoi, then a morphism f:(𝐗,𝒪)(𝐗,𝒪):𝑓𝐗𝒪superscript𝐗superscript𝒪f:({\mathbf{X}},{\mathcal{O}})\to({\mathbf{X}}^{\prime},{\mathcal{O}}^{\prime}) consists of a geometric morphism of topoi f:𝐗𝐗:𝑓𝐗superscript𝐗f:{\mathbf{X}}\to{\mathbf{X}}^{\prime} together with a ring homomorphism f#:𝒪f𝒪:subscript𝑓#superscript𝒪subscript𝑓𝒪f_{\#}:{\mathcal{O}}^{\prime}\to f_{*}{\mathcal{O}}, which then corresponds to a ring homomorphism f#:f𝒪𝒪:superscript𝑓#superscript𝑓superscript𝒪𝒪f^{\#}:f^{*}{\mathcal{O}}^{\prime}\to{\mathcal{O}} via the adjunction.

Now let (𝐗,𝒪)𝐗𝒪({\mathbf{X}},{\mathcal{O}}) and (𝐗,𝒪)superscript𝐗superscript𝒪({\mathbf{X}}^{\prime},{\mathcal{O}}^{\prime}) be ringed topoi with involutions (Λ,ν,λ)Λ𝜈𝜆(\Lambda,\nu,\lambda) and (Λ,ν,λ)superscriptΛsuperscript𝜈superscript𝜆(\Lambda^{\prime},\nu^{\prime},\lambda^{\prime}), respectively. Regarding ΛΛ\Lambda and ΛsuperscriptΛ\Lambda^{\prime} as geometric automorphisms of 𝐗𝐗{\mathbf{X}} and 𝐗superscript𝐗{\mathbf{X}}^{\prime}, see Remark 4.1.3, a morphism of ringed topoi with involution (𝐗,𝒪)(𝐗,𝒪)𝐗𝒪superscript𝐗superscript𝒪({\mathbf{X}},{\mathcal{O}})\to({\mathbf{X}}^{\prime},{\mathcal{O}}^{\prime}) should intuitively consist of a morphism of ringed topoi f𝑓f such that fΛ𝑓Λf\circ\Lambda is “equivalent” to ΛfsuperscriptΛ𝑓\Lambda^{\prime}\circ f. The specifications of this equivalence, which we now give, are somewhat technical.

Definition 4.2.1.

With the previous notation, a morphism of ringed topoi with involution (𝐗,𝒪)(𝐗,𝒪)𝐗𝒪superscript𝐗superscript𝒪({\mathbf{X}},{\mathcal{O}})\to({\mathbf{X}}^{\prime},{\mathcal{O}}^{\prime}) consists of a morphism of ringed topoi f:(𝐗,𝒪)(𝐗,𝒪):𝑓𝐗𝒪superscript𝐗superscript𝒪f:({\mathbf{X}},{\mathcal{O}})\to({\mathbf{X}}^{\prime},{\mathcal{O}}^{\prime}) together with natural isomorphisms α:fΛΛf:subscript𝛼subscript𝑓ΛsuperscriptΛsubscript𝑓\alpha_{*}:f_{*}\Lambda\Rightarrow\Lambda^{\prime}f_{*} and α:fΛΛf:superscript𝛼superscript𝑓superscriptΛΛsuperscript𝑓\alpha^{*}:f^{*}\Lambda^{\prime}\Rightarrow\Lambda f^{*} satisfying the following coherence conditions for all objects X𝑋X in 𝐗𝐗{\mathbf{X}} and Xsuperscript𝑋X^{\prime} in 𝐗superscript𝐗{\mathbf{X}}^{\prime}:

  1. (1)

    The following diagram, the columns of which are induced by αsuperscript𝛼\alpha^{*} and αsubscript𝛼\alpha_{*} and the rows of which are induced by the relevant adjunctions, commutes.

    Hom𝐗(ΛfX,X)subscriptHom𝐗Λsuperscript𝑓superscript𝑋𝑋\textstyle{\operatorname{Hom}_{{\mathbf{X}}}(\Lambda f^{*}X^{\prime},X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Hom𝐗(X,fΛX)subscriptHomsuperscript𝐗superscript𝑋subscript𝑓Λ𝑋\textstyle{\operatorname{Hom}_{{\mathbf{X}}^{\prime}}(X^{\prime},f_{*}\Lambda X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Hom𝐗(fΛX,X)subscriptHom𝐗superscript𝑓superscriptΛsuperscript𝑋𝑋\textstyle{\operatorname{Hom}_{{\mathbf{X}}}(f^{*}\Lambda^{\prime}X^{\prime},X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Hom𝐗(X,ΛfX)subscriptHomsuperscript𝐗superscript𝑋superscriptΛsubscript𝑓𝑋\textstyle{\operatorname{Hom}_{{\mathbf{X}}^{\prime}}(X^{\prime},\Lambda^{\prime}f_{*}X)}
  2. (2)

    fνX=νfXΛα,Xα,ΛXsubscript𝑓subscript𝜈𝑋subscriptsuperscript𝜈subscript𝑓𝑋superscriptΛsubscript𝛼𝑋subscript𝛼Λ𝑋f_{*}\nu_{X}=\nu^{\prime}_{f_{*}X}\circ\Lambda^{\prime}\alpha_{*,X}\circ\alpha_{*,\Lambda X}

  3. (3)

    Λf#λ=α,𝒪fλf#superscriptΛsubscript𝑓#superscript𝜆subscript𝛼𝒪subscript𝑓𝜆subscript𝑓#\Lambda^{\prime}f_{\#}\circ\lambda^{\prime}=\alpha_{*,{\mathcal{O}}}\circ f_{*}\lambda\circ f_{\#}

  4. (4)

    fνX=νfXΛαXαΛXsuperscript𝑓subscriptsuperscript𝜈superscript𝑋subscript𝜈superscript𝑓superscript𝑋Λsubscriptsuperscript𝛼superscript𝑋subscriptsuperscript𝛼superscriptΛsuperscript𝑋f^{*}\nu^{\prime}_{X^{\prime}}=\nu_{f^{*}X^{\prime}}\circ\Lambda\alpha^{*}_{X^{\prime}}\circ\alpha^{*}_{\Lambda^{\prime}X^{\prime}}

  5. (5)

    λf#=Λf#α𝒪fλ𝜆superscript𝑓#Λsuperscript𝑓#subscriptsuperscript𝛼superscript𝒪superscript𝑓superscript𝜆\lambda\circ f^{\#}=\Lambda f^{\#}\circ\alpha^{*}_{{\mathcal{O}}^{\prime}}\circ f^{*}\lambda^{\prime}

We say that f𝑓f is strict when α=idsubscript𝛼id\alpha_{*}=\mathrm{id} and α=idsuperscript𝛼id\alpha^{*}=\mathrm{id}, so that fΛ=Λfsubscript𝑓ΛsuperscriptΛsubscript𝑓f_{*}\Lambda=\Lambda^{\prime}f_{*} and Λf=fΛΛsuperscript𝑓superscript𝑓superscriptΛ\Lambda f^{*}=f^{*}\Lambda^{\prime}.

We call f𝑓f an equivalence when fsubscript𝑓f_{*} is an equivalence of categories and f#subscript𝑓#f_{\#} is an isomorphism, in which case the same holds for fsuperscript𝑓f^{*} and f#superscript𝑓#f^{\#}.

Remark 4.2.2.

Yoneda’s lemma and condition (1) imply that αsubscript𝛼\alpha_{*} and αsuperscript𝛼\alpha^{*} determine each other. Explicitly, this is given as

αX=εΛfX(f)fνfΛfXfΛα,ΛfXfΛfνfX1fΛηX.subscriptsuperscript𝛼superscript𝑋subscriptsuperscript𝜀𝑓Λsuperscript𝑓superscript𝑋superscript𝑓subscriptsuperscript𝜈subscript𝑓Λsuperscript𝑓superscript𝑋superscript𝑓superscriptΛsubscript𝛼Λsuperscript𝑓superscript𝑋superscript𝑓superscriptΛsubscript𝑓subscriptsuperscript𝜈1superscript𝑓superscript𝑋superscript𝑓superscriptΛsubscript𝜂superscript𝑋\alpha^{*}_{X^{\prime}}=\varepsilon^{(f)}_{\Lambda f^{*}X^{\prime}}\circ f^{*}\nu^{\prime}_{f_{*}\Lambda f^{*}X^{\prime}}\circ f^{*}\Lambda^{\prime}\alpha_{*,\Lambda f^{*}X^{\prime}}\circ f^{*}\Lambda^{\prime}f_{*}\nu^{-1}_{f^{*}X^{\prime}}\circ f^{*}\Lambda^{\prime}\eta_{X^{\prime}}\ .

Furthermore, provided (1) holds, conditions (2) and (4) are equivalent, and so are (3) and (5). Thus, in practice, it is enough to either specify α:fΛΛf:subscript𝛼subscript𝑓ΛsuperscriptΛsubscript𝑓\alpha_{*}:f_{*}\Lambda\Rightarrow\Lambda^{\prime}f_{*} and verify (2) and (3), or specify α:fΛΛf:superscript𝛼superscript𝑓superscriptΛΛsuperscript𝑓\alpha^{*}:f^{*}\Lambda^{\prime}\Rightarrow\Lambda f^{*} and verify (4) and (5).

In accordance with Remark 4.1.3, we will sometimes call morphisms of ringed topoi with involution C2subscript𝐶2C_{2}-equivariant morphisms.

We will usually suppress αsubscript𝛼\alpha_{*} and αsuperscript𝛼\alpha^{*} in computations, identifying fΛsubscript𝑓Λf_{*}\Lambda with ΛfsuperscriptΛsubscript𝑓\Lambda^{\prime}f_{*} and ΛfΛsuperscript𝑓\Lambda f^{*} with fΛsuperscript𝑓superscriptΛf^{*}\Lambda^{\prime}. The coherence conditions guarantee that this will not cause inconsistency or ambiguity.

Example 4.2.3.

Let (Λ,ν,λ)Λ𝜈𝜆(\Lambda,\nu,\lambda) be a weakly trivial involution of (𝐗,𝒪)𝐗𝒪({\mathbf{X}},{\mathcal{O}}) and let (Λ,ν,λ)superscriptΛsuperscript𝜈superscript𝜆(\Lambda^{\prime},\nu^{\prime},\lambda^{\prime}) be the trivial involution on (𝐗,𝒪)𝐗𝒪({\mathbf{X}},{\mathcal{O}}), see Definition 4.1.8. Then there is θ:Λid:𝜃Λid\theta:\Lambda\Rightarrow\mathrm{id} such that θ𝒪=λ1subscript𝜃𝒪superscript𝜆1\theta_{{\mathcal{O}}}=\lambda^{-1} and θXθΛX=νXsubscript𝜃𝑋subscript𝜃Λ𝑋subscript𝜈𝑋\theta_{X}\circ\theta_{\Lambda X}=\nu_{X} for all X𝑋X in 𝐗𝐗{\mathbf{X}}, and one can readily verify that (f,α,α):=(id(𝐗,𝒪),θ,θ1)assign𝑓subscript𝛼superscript𝛼subscriptid𝐗𝒪𝜃superscript𝜃1(f,\alpha_{*},\alpha^{*}):=(\mathrm{id}_{({\mathbf{X}},{\mathcal{O}})},\theta,\theta^{-1}) defines an equivalence from (𝐗,𝒪,Λ,ν,λ)𝐗𝒪Λ𝜈𝜆({\mathbf{X}},{\mathcal{O}},\Lambda,\nu,\lambda) to (𝐗,𝒪,Λ,ν,λ)𝐗𝒪superscriptΛsuperscript𝜈superscript𝜆({\mathbf{X}},{\mathcal{O}},\Lambda^{\prime},\nu^{\prime},\lambda^{\prime}), and that every such equivalence is of this form.

More generally, if (Λ,ν,λ)Λ𝜈𝜆(\Lambda,\nu,\lambda) is arbitrary and there exists an equivalence (f,α,α)𝑓subscript𝛼superscript𝛼(f,\alpha_{*},\alpha^{*}) from (𝐗,𝒪,Λ,ν,λ)𝐗𝒪Λ𝜈𝜆({\mathbf{X}},{\mathcal{O}},\Lambda,\nu,\lambda) to a ringed topos with a trivial involution, then (Λ,ν,λ)Λ𝜈𝜆(\Lambda,\nu,\lambda) is weakly trivial in the sense of Definition 4.1.8; take θX=εXfα,XεΛX1subscript𝜃𝑋subscript𝜀𝑋superscript𝑓subscript𝛼𝑋superscriptsubscript𝜀Λ𝑋1\theta_{X}=\varepsilon_{X}\circ f^{*}\alpha_{*,X}\circ\varepsilon_{\Lambda X}^{-1}.

If f:(𝐗1,𝒪1,Λ1,ν1,λ1)(𝐗2,𝒪2,Λ2,ν2,λ2):𝑓subscript𝐗1subscript𝒪1subscriptΛ1subscript𝜈1subscript𝜆1subscript𝐗2subscript𝒪2subscriptΛ2subscript𝜈2subscript𝜆2f:(\mathbf{X}_{1},\mathcal{O}_{1},\Lambda_{1},\nu_{1},\lambda_{1})\to(\mathbf{X}_{2},\mathcal{O}_{2},\Lambda_{2},\nu_{2},\lambda_{2}) is a morphism of ringed topoi with involution, and if A𝐴A is an Azumaya 𝒪2subscript𝒪2{\mathcal{O}}_{2}-algebra with a λ2subscript𝜆2\lambda_{2}-involution τ𝜏\tau, then fAf𝒪2𝒪1subscripttensor-productsuperscript𝑓subscript𝒪2superscript𝑓𝐴subscript𝒪1f^{*}A\otimes_{f^{*}{\mathcal{O}}_{2}}{\mathcal{O}}_{1} is an Azumaya algebra on (𝐗1,𝒪1)subscript𝐗1subscript𝒪1(\mathbf{X}_{1},{\mathcal{O}}_{1}) with a λ1subscript𝜆1\lambda_{1}-involution given by fτλ1tensor-productsuperscript𝑓𝜏subscript𝜆1f^{*}\tau\otimes\lambda_{1}. This induces transfer functors

Azn(𝒪2)Azn(𝒪1)andAzn(𝒪2,λ2)Azn(𝒪1,λ1).formulae-sequencesubscriptAz𝑛subscript𝒪2subscriptAz𝑛subscript𝒪1andsubscriptAz𝑛subscript𝒪2subscript𝜆2subscriptAz𝑛subscript𝒪1subscript𝜆1{\text{\bf Az}}_{n}({\mathcal{O}}_{2})\to{\text{\bf Az}}_{n}({\mathcal{O}}_{1})\qquad\text{and}\qquad{\text{\bf Az}}_{n}({\mathcal{O}}_{2},\lambda_{2})\to{\text{\bf Az}}_{n}({\mathcal{O}}_{1},\lambda_{1})\ .

We shall need a particular instance of these transfer maps later.

Example 4.2.4.

If X𝑋X is a complex variety with involution λ:XX:𝜆𝑋𝑋\lambda:X\to X, then the étale ringed topos of X𝑋X, denoted 𝐗étsubscript𝐗ét\mathbf{X}_{\text{\'{e}t}}, becomes a locally ringed topos with involution, as in Example 4.1.5. On the other hand, one may form the topological space X()𝑋X(\mathbb{C}), equipped with the analytic topology, which then has an associated site and consequently an associated topos XtopsubscriptXtop\text{\bf X}_{\mathrm{top}}. One may endow the topos XtopsubscriptXtop\text{\bf X}_{\mathrm{top}} with several different local ring objects depending on the kind of geometry one wishes to carry out. There is the sheaf \mathcal{H} of holomorphic functions, as set out in [grothendieck_techniques_1960], and there is also the sheaf 𝒞𝒞\mathcal{C} of continuous \mathbb{C}-valued functions.

We claim that (Xtop,𝒞)subscriptXtop𝒞(\text{\bf X}_{\mathrm{top}},\mathcal{C}) is a locally ringed topos and that there is a “realization” morphism (Xtop,𝒞)(Xét,𝒪X)subscriptXtop𝒞subscriptXétsubscript𝒪𝑋(\text{\bf X}_{\mathrm{top}},\mathcal{C})\to(\text{\bf X}_{\text{\'{e}t}},\mathcal{O}_{X}) of ringed topoi with involution. An outline of the argument follows.

For every complex variety U𝑈U, there is a unique, functorially-defined analytic topology on U()𝑈U(\mathbb{C}); this is established in [grothendieck_revetements_1971, Exp. XII]. The functor UU()maps-to𝑈𝑈U\mapsto U(\mathbb{C}) preserves finite limits. Moreover, if {UiX}iIsubscriptsubscript𝑈𝑖𝑋𝑖𝐼\{U_{i}\to X\}_{i\in I} is an étale covering of X𝑋X, then the family of maps {Ui()X()}iIsubscriptsubscript𝑈𝑖𝑋𝑖𝐼\{U_{i}(\mathbb{C})\to X(\mathbb{C})\}_{i\in I} is a jointly surjective family of local homeomorphisms, and may therefore be refined by a jointly surjective family {ViX()}iIsubscriptsubscript𝑉𝑖𝑋𝑖superscript𝐼\{V_{i}\to X(\mathbb{C})\}_{i\in I^{\prime}} of open inclusions. Since families of this form generate the usual Grothendieck topology on the topological space X()𝑋X(\mathbb{C}), it follows that there is a morphism of sites f:(X(),top)Xét:𝑓𝑋topsubscript𝑋étf:(X(\mathbb{C}),\mathrm{top})\to X_{\text{\'{e}t}}, and therefore a morphism of topoi, [artin_theorie_1972-1, III.1 and IV.4.9.4]. Complex realization may be applied to 𝔸1subscriptsuperscript𝔸1\mathbb{A}^{1}_{\mathbb{C}}, the representing object for 𝒪Xsubscript𝒪𝑋\mathcal{O}_{X}, to obtain \mathbb{C}, the representing object for 𝒞𝒞\mathcal{C} which is the local ring object on X()𝑋X(\mathbb{C}). Therefore, f𝑓f is a morphism of locally ringed topoi. It is routine to verify that since the involution on X𝑋X becomes the obvious involution on X()𝑋X(\mathbb{C}) after realization, the morphism f:𝐗top𝐗ét:𝑓subscript𝐗topsubscript𝐗étf:\mathbf{X}_{\mathrm{top}}\to\mathbf{X}_{\text{\'{e}t}} extends to a strict morphism of locally ringed topoi with involution. In this instance, all the “coherence” natural isomorphisms appearing are, in fact, identities.

4.3. Quotients by an Involution

Let λ=(Λ,ν,λ)𝜆Λ𝜈𝜆\lambda=(\Lambda,\nu,\lambda) be an involution of a locally ringed topos 𝐗𝐗{\mathbf{X}}. We would like to consider a quotient of 𝐗𝐗{\mathbf{X}} by the action of λ𝜆\lambda, or equivalently, by the (weak) C2subscript𝐶2C_{2}-action it induces. In general, however, it is difficult to define a specific quotient topos in a geometrically reasonable way. For example, if (𝐗,𝒪𝐗)=(Sh(Xét),𝒪X)𝐗subscript𝒪𝐗Shsubscript𝑋étsubscript𝒪𝑋(\mathbf{X},{\mathcal{O}}_{\mathbf{X}})=(\text{\bf Sh}(X_{\text{\'{e}t}}),{\mathcal{O}}_{X}) for a scheme X𝑋X admitting a C2subscript𝐶2C_{2}-action, then the étale ringed topoi of both the geometric quotient X/C2𝑋subscript𝐶2X/C_{2}, if exists, and the stack [X/C2]delimited-[]𝑋subscript𝐶2[X/C_{2}] may a priori serve as reasonable quotients of 𝐗𝐗{\mathbf{X}}.

We therefore ignore the problem of constructing or specifying a quotient of a locally ringed topos with involution and instead enumerate the properties that such a quotient should possess, declaring any locally ringed topos possessing these properties to be satisfactory.

To be precise, we ask for a locally ringed topos 𝐘𝐘{\mathbf{Y}}, endowed with the trivial involution, together with a C2subscript𝐶2C_{2}-equivariant morphism π:𝐗𝐘:𝜋𝐗𝐘\pi:{\mathbf{X}}\to{\mathbf{Y}} which satisfy certain axioms. Recall from Subsection 4.2 that the data of π𝜋\pi consists of a geometric morphism of topoi π=(π,π):𝐗𝐘:𝜋superscript𝜋subscript𝜋𝐗𝐘\pi=(\pi^{*},\pi_{*}):\mathbf{X}\to\mathbf{Y}, a ring homomorphism π#:𝒪𝐘π𝒪𝐗:subscript𝜋#subscript𝒪𝐘subscript𝜋subscript𝒪𝐗\pi_{\#}:{\mathcal{O}}_{\mathbf{Y}}\to\pi_{*}{\mathcal{O}}_{\mathbf{X}} (or equivalently, π#:π𝒪𝐘𝒪𝐗:superscript𝜋#superscript𝜋subscript𝒪𝐘subscript𝒪𝐗\pi^{\#}:\pi^{*}{\mathcal{O}}_{\mathbf{Y}}\to{\mathcal{O}}_{\mathbf{X}}) and natural transformations α:πΛπ:subscript𝛼subscript𝜋Λsubscript𝜋\alpha_{*}:\pi_{*}\Lambda\Rightarrow\pi_{*}, α:πΛπ:superscript𝛼superscript𝜋Λsuperscript𝜋\alpha^{*}:\pi^{*}\Rightarrow\Lambda\pi^{*} satisfying the relations of Definition 4.2.1. We will often suppress αsubscript𝛼\alpha_{*} and αsuperscript𝛼\alpha^{*}, identifying πΛsubscript𝜋Λ\pi_{*}\Lambda with πsubscript𝜋\pi_{*} and ΛπΛsuperscript𝜋\Lambda\pi^{*} with πsuperscript𝜋\pi^{*}. In fact, in many of our examples, αsubscript𝛼\alpha_{*} and αsuperscript𝛼\alpha^{*} will both be the identity.

Definition 4.3.1.

Let (𝐗,𝒪𝐗)𝐗subscript𝒪𝐗(\mathbf{X},{\mathcal{O}}_{{\mathbf{X}}}) be a locally ringed topos with involution λ=(Λ,ν,λ)𝜆Λ𝜈𝜆\lambda=(\Lambda,\nu,\lambda), let (𝐘,𝒪𝐘)𝐘subscript𝒪𝐘(\mathbf{Y},{\mathcal{O}}_{{\mathbf{Y}}}) be a locally ringed topos with a trivial involution, and let π:(𝐗,𝒪𝐗)(𝐘,𝒪𝐘):𝜋𝐗subscript𝒪𝐗𝐘subscript𝒪𝐘\pi:(\mathbf{X},{\mathcal{O}}_{{\mathbf{X}}})\to(\mathbf{Y},{\mathcal{O}}_{{\mathbf{Y}}}) be a C2subscript𝐶2C_{2}-equivariant morphism of ringed topoi. We say that π𝜋\pi is an exact quotient (of (𝐗,𝒪𝐗)𝐗subscript𝒪𝐗(\mathbf{X},{\mathcal{O}}_{{\mathbf{X}}}) by the given C2subscript𝐶2C_{2}-action) if

  1. (E1)

    π#:𝒪𝐘π𝒪𝐗:subscript𝜋#subscript𝒪𝐘subscript𝜋subscript𝒪𝐗\pi_{\#}:{\mathcal{O}}_{{\mathbf{Y}}}\to\pi_{*}{\mathcal{O}}_{{\mathbf{X}}} is the equalizer of πλ:π𝒪𝐗πΛ𝒪𝐗=π𝒪𝐗:subscript𝜋𝜆subscript𝜋subscript𝒪𝐗subscript𝜋Λsubscript𝒪𝐗subscript𝜋subscript𝒪𝐗\pi_{*}\lambda:\pi_{*}{\mathcal{O}}_{{\mathbf{X}}}\to\pi_{*}\Lambda{\mathcal{O}}_{{\mathbf{X}}}=\pi_{*}{\mathcal{O}}_{{\mathbf{X}}} and the identity map idπ𝒪𝐗subscriptidsubscript𝜋subscript𝒪𝐗\mathrm{id}_{\pi_{*}{\mathcal{O}}_{\mathbf{X}}},

  2. (E2)

    πsubscript𝜋\pi_{*} preserves epimorphisms.

Remark 4.3.2.

An exact quotient is in particular a morphism of locally ringed topoi, i.e., a morphism of ringed topoi π:𝐗𝐘:𝜋𝐗𝐘\pi:{\mathbf{X}}\to{\mathbf{Y}} satisfying the additional condition that 𝒪𝐘×𝒪𝐘superscriptsubscript𝒪𝐘subscript𝒪𝐘{{\mathcal{O}}_{\mathbf{Y}}^{\times}}\to{\mathcal{O}}_{\mathbf{Y}} is the pullback of π𝒪𝐗×π𝒪𝐗subscript𝜋superscriptsubscript𝒪𝐗subscript𝜋subscript𝒪𝐗{\pi_{*}{\mathcal{O}}_{\mathbf{X}}^{\times}}\to\pi_{*}{\mathcal{O}}_{\mathbf{X}} along π#subscript𝜋#\pi_{\#}. Indeed, given V𝐘𝑉𝐘V\in{\mathbf{Y}}, the V𝑉V-sections of the pullback consist of pairs (x,y)π𝒪𝐗×(V)×𝒪𝐘(V)𝑥𝑦subscript𝜋superscriptsubscript𝒪𝐗𝑉subscript𝒪𝐘𝑉(x,y)\in\pi_{*}{{\mathcal{O}}_{\mathbf{X}}^{\times}}(V)\times{\mathcal{O}}_{\mathbf{Y}}(V) with π#y=xsubscript𝜋#𝑦𝑥\pi_{\#}y=x in π𝒪𝐗(V)subscript𝜋subscript𝒪𝐗𝑉\pi_{*}{\mathcal{O}}_{\mathbf{X}}(V). By (E1), we have πλ(x)=xsubscript𝜋𝜆𝑥𝑥\pi_{*}\lambda(x)=x in 𝒪𝐗(V)subscript𝒪𝐗𝑉{\mathcal{O}}_{\mathbf{X}}(V). Since xπ𝒪𝐗×(V)=π𝒪𝐗(V)×𝑥subscript𝜋superscriptsubscript𝒪𝐗𝑉subscript𝜋subscript𝒪𝐗superscript𝑉x\in\pi_{*}{{\mathcal{O}}_{\mathbf{X}}^{\times}}(V)={\pi_{*}{\mathcal{O}}_{\mathbf{X}}(V)^{\times}}, this means that πλ(x1)=x1subscript𝜋𝜆superscript𝑥1superscript𝑥1\pi_{*}\lambda(x^{-1})=x^{-1}. Thus, again by (E1), there exists unique y𝒪𝐘(V)superscript𝑦subscript𝒪𝐘𝑉y^{\prime}\in{\mathcal{O}}_{\mathbf{Y}}(V) with π#y=x1subscript𝜋#superscript𝑦superscript𝑥1\pi_{\#}y^{\prime}=x^{-1}. In particular, π#(yy)=xx1=1subscript𝜋#𝑦superscript𝑦𝑥superscript𝑥11\pi_{\#}(yy^{\prime})=xx^{-1}=1 in π𝒪𝐗(V)subscript𝜋subscript𝒪𝐗𝑉\pi_{*}{\mathcal{O}}_{\mathbf{X}}(V). Since 𝒪𝐘subscript𝒪𝐘{\mathcal{O}}_{\mathbf{Y}} is a subobject of π𝒪𝐗subscript𝜋subscript𝒪𝐗\pi_{*}{\mathcal{O}}_{\mathbf{X}} via π#subscript𝜋#\pi_{\#} (again, by (E1)), this means that yy=1𝑦superscript𝑦1yy^{\prime}=1 in 𝒪𝐘(V)subscript𝒪𝐘𝑉{\mathcal{O}}_{\mathbf{Y}}(V), so y𝒪𝐘(V)×=𝒪𝐘×(V)𝑦subscript𝒪𝐘superscript𝑉superscriptsubscript𝒪𝐘𝑉y\in{{\mathcal{O}}_{\mathbf{Y}}(V)^{\times}}={{\mathcal{O}}_{\mathbf{Y}}^{\times}}(V). As x=π#y𝑥subscript𝜋#𝑦x=\pi_{\#}y, it follows that (x,y)𝑥𝑦(x,y) is the image of a (necessarily unique) V𝑉V-section of 𝒪𝐘×superscriptsubscript𝒪𝐘{{\mathcal{O}}_{\mathbf{Y}}^{\times}} under the natural map 𝒪𝐘×π𝒪𝐗××π𝒪𝐗𝒪𝐘superscriptsubscript𝒪𝐘subscriptsubscript𝜋subscript𝒪𝐗subscript𝜋superscriptsubscript𝒪𝐗subscript𝒪𝐘{{\mathcal{O}}_{\mathbf{Y}}^{\times}}\to\pi_{*}{{\mathcal{O}}_{\mathbf{X}}^{\times}}\times_{\pi_{*}{\mathcal{O}}_{\mathbf{X}}}{\mathcal{O}}_{\mathbf{Y}}, as required.

The name “exact” comes from condition (E2), which implies in particular that πsubscript𝜋\pi_{*} preserves exact sequences of groups. We shall see below that this condition is critical for transferring cohomological data from 𝐗𝐗{\mathbf{X}} to 𝐘𝐘{\mathbf{Y}}. Condition (E1) informally means that 𝒪𝐘subscript𝒪𝐘{\mathcal{O}}_{\mathbf{Y}} behaves as one would expect from the subring of 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}} fixed by λ𝜆\lambda — such an object cannot be defined in 𝐗𝐗{\mathbf{X}} because the source and target of λ𝜆\lambda are not, in general, canonically isomorphic.

To motivate Definition 4.3.1, we now give two fundamental examples of exact quotients. However, in order not to digress, we postpone the proof of their exactness to Subsection 4.4, where further examples and non-examples are exhibited.

Example 4.3.3.

Let X𝑋X be a scheme and let λ:XX:𝜆𝑋𝑋\lambda:X\to X be an involution. A morphism of schemes π:XY:𝜋𝑋𝑌\pi:X\to Y is called a good quotient of X𝑋X relative to the action of C2={1,λ}subscript𝐶21𝜆C_{2}=\{1,\lambda\} if π𝜋\pi is affine, C2subscript𝐶2C_{2}-invariant, and π#:𝒪Yπ𝒪X:subscript𝜋#subscript𝒪𝑌subscript𝜋subscript𝒪𝑋\pi_{\#}:{\mathcal{O}}_{Y}\to\pi_{*}{\mathcal{O}}_{X} defines an isomorphism of 𝒪Ysubscript𝒪𝑌{\mathcal{O}}_{Y} with (π𝒪X)C2superscriptsubscript𝜋subscript𝒪𝑋subscript𝐶2(\pi_{*}{\mathcal{O}}_{X})^{C_{2}}. By [grothendieck_revetements_1971, Prp. V.1.3] and the going-up theorem, these conditions imply that π𝜋\pi is universally surjective and so this agrees with the more general definition in [de_jong_stacks_2017, Tag 04AB]. A good C2subscript𝐶2C_{2}-quotient of X𝑋X exists if and only if every C2subscript𝐶2C_{2}-orbit in X𝑋X is contained in an affine open subscheme, in which case it is also a categorical quotient in the category of schemes, hence unique up to isomorphism [grothendieck_revetements_1971, Prps. V.1.3, V.1.8].

Let (𝐗,𝒪𝐗)=(Sh(Xét),𝒪X)𝐗subscript𝒪𝐗Shsubscript𝑋étsubscript𝒪𝑋({\mathbf{X}},{\mathcal{O}}_{\mathbf{X}})=(\text{\bf Sh}(X_{\text{\'{e}t}}),{\mathcal{O}}_{X}), and let λ=(Λ,ν,λ)𝜆Λ𝜈𝜆\lambda=(\Lambda,\nu,\lambda) be the involution of (𝐗,𝒪𝐗)𝐗subscript𝒪𝐗({\mathbf{X}},{\mathcal{O}}_{\mathbf{X}}) induced by λ:XX:𝜆𝑋𝑋\lambda:X\to X, see Example 4.1.5. Given a good C2subscript𝐶2C_{2}-quotient, π:XY:𝜋𝑋𝑌\pi:X\to Y, we define an exact quotient π:(𝐗,𝒪𝐗)(𝐘,𝒪𝐘):𝜋𝐗subscript𝒪𝐗𝐘subscript𝒪𝐘\pi:({\mathbf{X}},{\mathcal{O}}_{\mathbf{X}})\to({\mathbf{Y}},{\mathcal{O}}_{\mathbf{Y}}) relative to λ𝜆\lambda by taking (𝐘,𝒪𝐘)=(Sh(Yét),𝒪Y)𝐘subscript𝒪𝐘Shsubscript𝑌étsubscript𝒪𝑌({\mathbf{Y}},{\mathcal{O}}_{\mathbf{Y}})=(\text{\bf Sh}(Y_{\text{\'{e}t}}),{\mathcal{O}}_{Y}), letting π=(π,π):Sh(Xét)Sh(Yét):𝜋superscript𝜋subscript𝜋Shsubscript𝑋étShsubscript𝑌ét\pi=(\pi^{*},\pi_{*}):\text{\bf Sh}(X_{\text{\'{e}t}})\to\text{\bf Sh}(Y_{\text{\'{e}t}}) and defining π#:𝒪𝐘π𝒪𝐗:subscript𝜋#subscript𝒪𝐘subscript𝜋subscript𝒪𝐗\pi_{\#}:{\mathcal{O}}_{\mathbf{Y}}\to\pi_{*}{\mathcal{O}}_{\mathbf{X}} to be the canonical extension of π#:𝒪Yπ𝒪X:subscript𝜋#subscript𝒪𝑌subscript𝜋subscript𝒪𝑋\pi_{\#}:{\mathcal{O}}_{Y}\to\pi_{*}{\mathcal{O}}_{X} in Sh(XZar)Shsubscript𝑋Zar\text{\bf Sh}(X_{\operatorname{Zar}}) to the corresponding ring objects in Sh(Xét)Shsubscript𝑋ét\text{\bf Sh}(X_{\text{\'{e}t}}). The suppressed natural transformations αsubscript𝛼\alpha_{*} and αsuperscript𝛼\alpha^{*} are both the identity.

Example 4.3.4.

Let X𝑋X be a Hausdorff topological space, let λ:XX:𝜆𝑋𝑋\lambda:X\to X be a continuous involution, let Y=X/{1,λ}𝑌𝑋1𝜆Y=X/\{1,\lambda\} and let π:XY:𝜋𝑋𝑌\pi:X\to Y be the quotient map. Let (𝐗,𝒪𝐗)=(Sh(X),𝒞(X,))𝐗subscript𝒪𝐗Sh𝑋𝒞𝑋({\mathbf{X}},{\mathcal{O}}_{\mathbf{X}})=(\text{\bf Sh}(X),{\mathcal{C}}(X,\mathbb{C})) and let λ=(Λ,ν,λ)𝜆Λ𝜈𝜆\lambda=(\Lambda,\nu,\lambda) be the involution induced by λ:XX:𝜆𝑋𝑋\lambda:X\to X, see Example 4.1.6. We define an exact quotient π:(𝐗,𝒪𝐗)(𝐘,𝒪𝐘):𝜋𝐗subscript𝒪𝐗𝐘subscript𝒪𝐘\pi:({\mathbf{X}},{\mathcal{O}}_{\mathbf{X}})\to({\mathbf{Y}},{\mathcal{O}}_{\mathbf{Y}}) relative to λ𝜆\lambda by taking (𝐘,𝒪𝐘)=(Sh(Y),𝒞(Y,))𝐘subscript𝒪𝐘Sh𝑌𝒞𝑌({\mathbf{Y}},{\mathcal{O}}_{\mathbf{Y}})=(\text{\bf Sh}(Y),{\mathcal{C}}(Y,\mathbb{C})), letting π=(π,π):Sh(X)Sh(Y):𝜋superscript𝜋subscript𝜋Sh𝑋Sh𝑌\pi=(\pi^{*},\pi_{*}):\text{\bf Sh}(X)\to\text{\bf Sh}(Y) be the geometric morphism induced by π:XY:𝜋𝑋𝑌\pi:X\to Y, and defining π#:𝒞(Y,)π𝒞(X,):subscript𝜋#𝒞𝑌subscript𝜋𝒞𝑋\pi_{\#}:{\mathcal{C}}(Y,\mathbb{C})\to\pi_{*}{\mathcal{C}}(X,\mathbb{C}) to be the morphism sending a section f𝒞(U,)𝑓𝒞𝑈f\in{\mathcal{C}}(U,\mathbb{C}) to fπ𝒞(π1(U),)=π𝒞(X,)(U)𝑓𝜋𝒞superscript𝜋1𝑈subscript𝜋𝒞𝑋𝑈f\circ\pi\in{\mathcal{C}}(\pi^{-1}(U),\mathbb{C})=\pi_{*}{\mathcal{C}}(X,\mathbb{C})(U). Again, the suppressed natural transformations αsubscript𝛼\alpha_{*} and αsuperscript𝛼\alpha^{*} are both the identity.

We also record the following trivial example.

Example 4.3.5.

Suppose that the involution λ=(Λ,ν,λ)𝜆Λ𝜈𝜆\lambda=(\Lambda,\nu,\lambda) on (𝐗,𝒪𝐗)𝐗subscript𝒪𝐗({\mathbf{X}},{\mathcal{O}}_{\mathbf{X}}) is weakly trivial, namely, there is a natural isomorphism θ:Λid:𝜃Λid\theta:\Lambda\Rightarrow\mathrm{id} such that θXθΛX=νXsubscript𝜃𝑋subscript𝜃Λ𝑋subscript𝜈𝑋\theta_{X}\circ\theta_{\Lambda X}=\nu_{X}, and λ=θ𝒪𝐗1𝜆subscriptsuperscript𝜃1subscript𝒪𝐗\lambda=\theta^{-1}_{{\mathcal{O}}_{\mathbf{X}}}. Then the identity morphism id:(𝐗,𝒪𝐗)(𝐗,𝒪𝐗):id𝐗subscript𝒪𝐗𝐗subscript𝒪𝐗\mathrm{id}:({\mathbf{X}},{\mathcal{O}}_{\mathbf{X}})\to({\mathbf{X}},{\mathcal{O}}_{\mathbf{X}}) defines an exact quotient upon taking α=θsubscript𝛼𝜃\alpha_{*}=\theta and α=θ1superscript𝛼superscript𝜃1\alpha^{*}=\theta^{-1}. We call it the trivial quotient of (𝐗,𝒪𝐗)𝐗subscript𝒪𝐗({\mathbf{X}},{\mathcal{O}}_{\mathbf{X}}).

More generally, an arbitrary exact C2subscript𝐶2C_{2}-quotient π:𝐗𝐘:𝜋𝐗𝐘\pi:{\mathbf{X}}\to{\mathbf{Y}} will be called trivial when π𝜋\pi is an equivalence of ringed topoi with involution. As noted in Example 4.2.3, such a quotient can only exist when the involution of 𝐗𝐗{\mathbf{X}} is weakly trivial.

We shall see below (Remark 4.5.9) that a locally ringed topos with involution may admit several non-equivalent exact quotients.

We turn to establish some properties of exact quotients that will arise in the sequel. The most crucial of these will be the fact that when π:𝐗𝐘:𝜋𝐗𝐘\pi:{\mathbf{X}}\to{\mathbf{Y}} is an exact C2subscript𝐶2C_{2}-quotient, πsubscript𝜋\pi_{*} induces an equivalence between the Azumaya 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}}-algebras and the Azumaya π𝒪𝐗subscript𝜋subscript𝒪𝐗\pi_{*}{\mathcal{O}}_{\mathbf{X}}-algebras, and similarly for Azumaya algebras with a λ𝜆\lambda-involution.

The following theorem is a consequence of condition (E2).

Theorem 4.3.6.

Let π:𝐗𝐘:𝜋𝐗𝐘\pi:\mathbf{X}\to\mathbf{Y} be a geometric morphism of topoi such that πsubscript𝜋\pi_{*} preserves epimorphisms, and let G𝐺G be a group in 𝐗𝐗\mathbf{X}. Then:

  1. (i)

    πsubscript𝜋\pi_{*} induces an equivalence of categories Tors(𝐗,G)Tors(𝐘,πG)Tors𝐗𝐺Tors𝐘subscript𝜋𝐺\text{\bf Tors}(\mathbf{X},G)\to\text{\bf Tors}(\mathbf{Y},\pi_{*}G).

  2. (ii)

    There is a canonical isomorphism Hi(𝐘,πG)Hi(𝐗,G)superscriptH𝑖𝐘subscript𝜋𝐺superscriptH𝑖𝐗𝐺\mathrm{H}^{i}(\mathbf{Y},\pi_{*}G)\cong\mathrm{H}^{i}(\mathbf{X},G) when i=0,1𝑖01i=0,1, and for all i0𝑖0i\geq 0 when G𝐺G is abelian. For i=0𝑖0i=0, this is the canonical isomorphism H0(𝐘,πG)=H0(𝐗,G)superscriptH0𝐘subscript𝜋𝐺superscriptH0𝐗𝐺\mathrm{H}^{0}({\mathbf{Y}},\pi_{*}G)=\mathrm{H}^{0}({\mathbf{X}},G). For i=1𝑖1i=1, this isomorphism agrees with the one induced by (i) and Proposition 2.4.2(i).

  3. (iii)

    If 0GGG′′00superscript𝐺𝐺superscript𝐺′′00\to G^{\prime}\to G\to G^{\prime\prime}\to 0 is a short exact sequence of abelian groups then the isomorphism of (ii) gives rise to an isomorphism between the cohomology long exact sequence of GGG′′superscript𝐺𝐺superscript𝐺′′G^{\prime}\to G\to G^{\prime\prime} and the cohomology long exact sequence of πGπGπG′′subscript𝜋superscript𝐺subscript𝜋𝐺subscript𝜋superscript𝐺′′\pi_{*}G^{\prime}\to\pi_{*}G\to\pi_{*}G^{\prime\prime}. The same holds for the truncated long exact sequence of parts (ii) and (iii) of Proposition 2.4.2 when G,G,G′′superscript𝐺𝐺superscript𝐺′′G^{\prime},G,G^{\prime\prime} are not assumed to be abelian.

Proof.
  1. (i)

    We treat the topos as a site in its own canonical topology. In this language, [giraud_cohomologie_1971]*Chap. V, Sect. 3.1.1.1 says that πsubscript𝜋\pi_{*} induces an equivalence Tors(𝐗,G)𝐘Tors(𝐘,πG)Torssuperscript𝐗𝐺𝐘Tors𝐘subscript𝜋𝐺\text{\bf Tors}(\mathbf{X},G)^{\mathbf{Y}}\to\text{\bf Tors}(\mathbf{Y},\pi_{*}G), where the source is the category of G𝐺G-torsors P𝑃P for which there exists a covering U𝐘𝑈subscript𝐘U\to*_{\mathbf{Y}} such that PπUGπUsubscript𝑃superscript𝜋𝑈subscript𝐺superscript𝜋𝑈P_{\pi^{*}U}\cong G_{\pi^{*}U}. We claim that this applies to all G𝐺G-torsors, and so πsubscript𝜋\pi_{*} induces an equivalence Tors(𝐗,G)Tors(𝐘,πG)Tors𝐗𝐺Tors𝐘subscript𝜋𝐺\text{\bf Tors}(\mathbf{X},G)\to\text{\bf Tors}(\mathbf{Y},\pi_{*}G).

    Since a G𝐺G-torsor P𝑃P is trivialized by itself, it is also trivialized by any object mapping to P𝑃P, for instance by ππPsuperscript𝜋subscript𝜋𝑃\pi^{*}\pi_{*}P. It is therefore sufficient to show that the map πP𝐘subscript𝜋𝑃subscript𝐘\pi_{*}P\to*_{{\mathbf{Y}}} is an epimorphism. Since P𝐗𝑃subscript𝐗P\to*_{{\mathbf{X}}} is an epimorphism, our assumption implies that πPπ(𝐗)=𝐘subscript𝜋𝑃subscript𝜋subscript𝐗subscript𝐘\pi_{*}P\to\pi_{*}(*_{\mathbf{X}})=*_{{\mathbf{Y}}} is also an epimorphism, so the claim is verified.

  2. (ii)

    Suppose first that G𝐺G is abelian. The fact that πsubscript𝜋\pi_{*} is exact implies that the family of functors {GHi(𝐘,πG)}i0subscriptmaps-to𝐺superscriptH𝑖𝐘subscript𝜋𝐺𝑖0\{G\mapsto\mathrm{H}^{i}({\mathbf{Y}},\pi_{*}G)\}_{i\geq 0} from abelian groups in 𝐗𝐗{\mathbf{X}} to abelian groups forms a δ𝛿\delta-functor. Thus, the universality of derived functors implies that the canonical isomorphism H0(𝐗,G)H0(𝐘,πG)similar-tosuperscriptH0𝐗𝐺superscriptH0𝐘subscript𝜋𝐺\mathrm{H}^{0}({\mathbf{X}},G)\xrightarrow{\sim}\mathrm{H}^{0}({\mathbf{Y}},\pi_{*}G) gives rise to a unique natural transformation Hi(𝐗,G)Hi(𝐗,πG)superscriptH𝑖𝐗𝐺superscriptH𝑖𝐗subscript𝜋𝐺\mathrm{H}^{i}({\mathbf{X}},G)\to\mathrm{H}^{i}({\mathbf{X}},\pi_{*}G) for any abelian G𝐺G. Since πsubscript𝜋\pi_{*} takes injective abelian groups to injective abelian groups, {GHi(𝐘,πG)}i0subscriptmaps-to𝐺superscriptH𝑖𝐘subscript𝜋𝐺𝑖0\{G\mapsto\mathrm{H}^{i}({\mathbf{Y}},\pi_{*}G)\}_{i\geq 0} is an effaceable δ𝛿\delta-functor, hence universal. Applying the universality of the latter to the natural isomorphism H0(𝐘,πG)H0(𝐗,G)similar-tosuperscriptH0𝐘subscript𝜋𝐺superscriptH0𝐗𝐺\mathrm{H}^{0}({\mathbf{Y}},\pi_{*}G)\xrightarrow{\sim}\mathrm{H}^{0}({\mathbf{X}},G) implies that Hi(𝐗,G)Hi(𝐗,πG)superscriptH𝑖𝐗𝐺superscriptH𝑖𝐗subscript𝜋𝐺\mathrm{H}^{i}({\mathbf{X}},G)\to\mathrm{H}^{i}({\mathbf{X}},\pi_{*}G) is an isomorphism.

    We note that if we use Verdier’s Theorem, see Subsection 2.3, to describe Hi(𝐗,G)superscriptH𝑖𝐗𝐺\mathrm{H}^{i}({\mathbf{X}},G) and Hi(𝐘,πG)superscriptH𝑖𝐘subscript𝜋𝐺\mathrm{H}^{i}({\mathbf{Y}},\pi_{*}G), then the isomorphism is given by sending the cohomology class represented by gZi(U,G)𝑔superscript𝑍𝑖subscript𝑈𝐺g\in Z^{i}(U_{\bullet},G) to the cohomology class represented by πgZi(πU,πG)subscript𝜋𝑔superscript𝑍𝑖subscript𝜋subscript𝑈subscript𝜋𝐺\pi_{*}g\in Z^{i}(\pi_{*}U_{\bullet},\pi_{*}G). Notice that πUsubscript𝜋subscript𝑈\pi_{*}U_{\bullet}, which is just πU:𝚫𝐘:subscript𝜋subscript𝑈𝚫𝐘\pi_{*}\circ U_{\bullet}:{\bm{\Delta}}\to{\mathbf{Y}}, is a hypercovering since πsubscript𝜋\pi_{*} preserves epimorphisms and commutes with cosknsubscriptcosk𝑛{\mathrm{cosk}}_{n} for all n𝑛n. This isomorphism coincides with the one in the previous paragraph because they coincide on the 00-th cohomology.

    When i=1𝑖1i=1, the map we have just described is defined for an arbitrary group G𝐺G, and we take it to be the canonical morphism H1(𝐗,G)H1(𝐘,πG)superscriptH1𝐗𝐺superscriptH1𝐘subscript𝜋𝐺\mathrm{H}^{1}({\mathbf{X}},G)\to\mathrm{H}^{1}({\mathbf{Y}},\pi_{*}G). The construction in the proof of Proposition 2.4.2(i) implies that this map agrees with the one induced by (i) and Proposition 2.4.2(i), and thus H1(𝐗,G)H1(𝐘,πG)superscriptH1𝐗𝐺superscriptH1𝐘subscript𝜋𝐺\mathrm{H}^{1}({\mathbf{X}},G)\to\mathrm{H}^{1}({\mathbf{Y}},\pi_{*}G) is an isomorphism.

  3. (iii)

    In the abelian case, this follows from the argument given in (ii), which shows that {GHi(𝐗,G)}i0subscriptmaps-to𝐺superscriptH𝑖𝐗𝐺𝑖0\{G\mapsto\mathrm{H}^{i}({\mathbf{X}},G)\}_{i\geq 0} and {GHi(𝐘,πG)}i0subscriptmaps-to𝐺superscriptH𝑖𝐘subscript𝜋𝐺𝑖0\{G\mapsto\mathrm{H}^{i}({\mathbf{Y}},\pi_{*}G)\}_{i\geq 0} are isomorphic δ𝛿\delta-functors. In the nonabelian case, this follows from the proof of Proposition 2.4.2, parts (ii) and (iii). ∎

Remark 4.3.7.

When π:𝐗𝐘:𝜋𝐗𝐘\pi:{\mathbf{X}}\to{\mathbf{Y}} is a geometric morphism of topoi, with no further assumptions, one still has natural transformations Hi(𝐘,πG)Hi(𝐗,G)superscriptH𝑖𝐘subscript𝜋𝐺superscriptH𝑖𝐗𝐺\mathrm{H}^{i}({\mathbf{Y}},\pi_{*}G)\to\mathrm{H}^{i}({\mathbf{X}},G) for all G𝐺G abelian and i0𝑖0i\geq 0, or G𝐺G non-abelian and i=0,1𝑖01i=0,1, and if πsubscript𝜋\pi_{*} preserves epimorphisms, they coincide with the inverses of the isomorphisms of Theorem 4.3.6. In the abelian case, the construction is given as follows: Using the exactness of πsuperscript𝜋\pi^{*}, one finds that the canonical map H0(𝐘,A)H0(𝐗,πA)superscriptH0𝐘𝐴superscriptH0𝐗superscript𝜋𝐴\mathrm{H}^{0}({\mathbf{Y}},A)\to\mathrm{H}^{0}({\mathbf{X}},\pi^{*}A) gives rise to natural transformations Hi(𝐘,A)Hi(𝐗,πA)superscriptH𝑖𝐘𝐴superscriptH𝑖𝐗superscript𝜋𝐴\mathrm{H}^{i}({\mathbf{Y}},A)\to\mathrm{H}^{i}({\mathbf{X}},\pi^{*}A). Taking A=πG𝐴subscript𝜋𝐺A=\pi_{*}G and composing with the map Hi(𝐗,ππG)Hi(𝐗,G)superscriptH𝑖𝐗superscript𝜋subscript𝜋𝐺superscriptH𝑖𝐗𝐺\mathrm{H}^{i}({\mathbf{X}},\pi^{*}\pi_{*}G)\to\mathrm{H}^{i}({\mathbf{X}},G), induced by the counit ππGGsuperscript𝜋subscript𝜋𝐺𝐺\pi^{*}\pi_{*}G\to G, one obtains a natural transformation Hi(𝐘,πG)Hi(𝐗,G)superscriptH𝑖𝐘subscript𝜋𝐺superscriptH𝑖𝐗𝐺\mathrm{H}^{i}({\mathbf{Y}},\pi_{*}G)\to\mathrm{H}^{i}({\mathbf{X}},G). This map can be written explicitly on the level of cocycles, using Verdier’s Theorem, and be adapted to the non-abelian case when i=0,1𝑖01i=0,1.

Henceforth, let π:𝐗𝐘:𝜋𝐗𝐘\pi:{\mathbf{X}}\to{\mathbf{Y}} be an exact quotient relative to an involution λ=(Λ,ν,λ)𝜆Λ𝜈𝜆\lambda=(\Lambda,\nu,\lambda) on 𝐗𝐗{\mathbf{X}}. We write

R=π𝒪𝐗andS=𝒪𝐘formulae-sequence𝑅subscript𝜋subscript𝒪𝐗and𝑆subscript𝒪𝐘R=\pi_{*}{\mathcal{O}}_{{\mathbf{X}}}\qquad\text{and}\qquad S={\mathcal{O}}_{\mathbf{Y}}

for brevity, and, abusing the notation, we let λ:RR:𝜆𝑅𝑅\lambda:R\to R denote the involution πλ:π𝒪𝐗πΛ𝒪𝐗:subscript𝜋𝜆subscript𝜋subscript𝒪𝐗subscript𝜋Λsubscript𝒪𝐗\pi_{*}\lambda:\pi_{*}{\mathcal{O}}_{{\mathbf{X}}}\to\pi_{*}\Lambda{\mathcal{O}}_{{\mathbf{X}}}.

We shall use the following lemma freely to identify πGLn(𝒪𝐗)subscript𝜋subscriptGL𝑛subscript𝒪𝐗\pi_{*}\operatorname{GL}_{n}({\mathcal{O}}_{\mathbf{X}}) with GLn(R)subscriptGL𝑛𝑅\operatorname{GL}_{n}(R) and πPGLn(𝒪𝐗)subscript𝜋subscriptPGL𝑛subscript𝒪𝐗\pi_{*}\operatorname{PGL}_{n}({\mathcal{O}}_{\mathbf{X}}) with PGLn(R)subscriptPGL𝑛𝑅\operatorname{PGL}_{n}(R).

Lemma 4.3.8.

For all n𝑛n\in\mathbb{N}, there are canonical isomorphisms πMn×n(𝒪𝐗)Mn×n(R)subscript𝜋subscriptM𝑛𝑛subscript𝒪𝐗subscriptM𝑛𝑛𝑅\pi_{*}\mathrm{M}_{n\times n}({\mathcal{O}}_{\mathbf{X}})\cong\mathrm{M}_{n\times n}(R), πGLn(𝒪𝐗)GLn(R)subscript𝜋subscriptGL𝑛subscript𝒪𝐗subscriptGL𝑛𝑅\pi_{*}\operatorname{GL}_{n}({\mathcal{O}}_{{\mathbf{X}}})\cong\operatorname{GL}_{n}(R) and πPGLn(𝒪𝐗)PGLn(R)subscript𝜋subscriptPGL𝑛subscript𝒪𝐗subscriptPGL𝑛𝑅\pi_{*}\operatorname{PGL}_{n}({\mathcal{O}}_{{\mathbf{X}}})\cong\operatorname{PGL}_{n}(R).

Proof.

Let U𝐘𝑈𝐘U\in{\mathbf{Y}}. Thanks to the adjunction between πsuperscript𝜋\pi^{*} and πsubscript𝜋\pi_{*}, we have a natural isomorphism πMn×n(𝒪𝐗)(U)Mn×n(𝒪𝐗)(πU)=Mn×n(𝒪𝐗(πU))Mn×n(π𝒪𝐗(U))subscript𝜋subscriptM𝑛𝑛subscript𝒪𝐗𝑈subscriptM𝑛𝑛subscript𝒪𝐗superscript𝜋𝑈subscriptM𝑛𝑛subscript𝒪𝐗superscript𝜋𝑈subscriptM𝑛𝑛subscript𝜋subscript𝒪𝐗𝑈\pi_{*}\mathrm{M}_{n\times n}({\mathcal{O}}_{\mathbf{X}})(U)\cong\mathrm{M}_{n\times n}({\mathcal{O}}_{\mathbf{X}})(\pi^{*}U)=\mathrm{M}_{n\times n}({\mathcal{O}}_{\mathbf{X}}(\pi^{*}U))\cong\mathrm{M}_{n\times n}(\pi_{*}{\mathcal{O}}_{\mathbf{X}}(U)). This establishes the first isomorphism.

The second isomorphism is obtained in the same manner.

The last isomorphism is deduced from the following ladder of short exact sequences

11\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}R×superscript𝑅\textstyle{R^{\times}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}\scriptstyle{\cong}GLn(R)subscriptGL𝑛𝑅\textstyle{\operatorname{GL}_{n}(R)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}\scriptstyle{\cong}PGLn(R)subscriptPGL𝑛𝑅\textstyle{\operatorname{PGL}_{n}(R)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}\scriptstyle{\cong}11\textstyle{1}11\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}π(𝒪𝐗×)subscript𝜋superscriptsubscript𝒪𝐗\textstyle{\pi_{*}({\mathcal{O}}_{\mathbf{X}}^{\times})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}π(GLn(𝒪𝐗))subscript𝜋subscriptGL𝑛subscript𝒪𝐗\textstyle{\pi_{*}(\operatorname{GL}_{n}({\mathcal{O}}_{\mathbf{X}}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}πPGLn(𝒪𝐗)subscript𝜋subscriptPGL𝑛subscript𝒪𝐗\textstyle{\pi_{*}\operatorname{PGL}_{n}({\mathcal{O}}_{\mathbf{X}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}1.1\textstyle{1.}

Here, the left and middle isomorphisms follow from the previous paragraph, and the bottom row is exact since πsubscript𝜋\pi_{*} preserves epimorphisms. ∎

We shall use the following lemma to identify PGLn(R)subscriptPGL𝑛𝑅\operatorname{PGL}_{n}(R) with 𝒜utR-alg(Mn×n(R))𝒜𝑢subscript𝑡R-algsubscriptM𝑛𝑛𝑅\mathcal{A}ut_{\text{$R$-alg}}(\mathrm{M}_{n\times n}(R)) henceforth.

Lemma 4.3.9.

The canonical group homomorphism PGLn(R)𝒜utR-alg(Mn×n(R))subscriptPGL𝑛𝑅𝒜𝑢subscript𝑡R-algsubscriptM𝑛𝑛𝑅\operatorname{PGL}_{n}(R)\to\mathcal{A}ut_{\text{$R$-alg}}(\mathrm{M}_{n\times n}(R)) is an isomorphism.

The lemma would follow immediately from the discussion in Subsection 2.5 if R𝑅R were a local ring object, but this is not the case in general. Using Theorem 3.3.8, we shall see that R𝑅R functions as a “semilocal” ring object and therefore the isomorphism still holds.

Proof.

We need to show that for all U𝐘𝑈𝐘U\in{\mathbf{Y}}, any RUsubscript𝑅𝑈R_{U}-automorphism ψ𝜓\psi of Mn×n(RU)subscriptM𝑛𝑛subscript𝑅𝑈\mathrm{M}_{n\times n}(R_{U}) becomes an inner automorphism after passing to a covering VU𝑉𝑈V\to U. Let 𝔭SpecS(U)𝔭Spec𝑆𝑈{\mathfrak{p}}\in\operatorname{Spec}S(U). Then B:=S(U)𝔭assign𝐵𝑆subscript𝑈𝔭B:=S(U)_{\mathfrak{p}} is the λ𝜆\lambda-fixed subring of A:=R(U)𝔭assign𝐴𝑅subscript𝑈𝔭A:=R(U)_{\mathfrak{p}}. By Theorem 3.3.8, A𝐴A is semilocal. It is then well known that ψAsubscript𝜓𝐴\psi_{A} is an inner automorphism, [knus_quadratic_1991, III.§5.2]. Write ψA(x)=a𝔭xa𝔭1subscript𝜓𝐴𝑥subscript𝑎𝔭𝑥superscriptsubscript𝑎𝔭1\psi_{A}(x)=a_{\mathfrak{p}}xa_{\mathfrak{p}}^{-1} for some a𝔭GLn(A)subscript𝑎𝔭subscriptGL𝑛𝐴a_{\mathfrak{p}}\in\operatorname{GL}_{n}(A). There exists f𝔭S(U)𝔭subscript𝑓𝔭𝑆𝑈𝔭f_{\mathfrak{p}}\in S(U)-{\mathfrak{p}} such that a𝔭subscript𝑎𝔭a_{\mathfrak{p}} is the image of an element in GLn(R(U)f𝔭)subscriptGL𝑛𝑅subscript𝑈subscript𝑓𝔭\operatorname{GL}_{n}(R(U)_{f_{\mathfrak{p}}}), also denoted a𝔭subscript𝑎𝔭a_{\mathfrak{p}}, and such that ψR(U)f𝔭subscript𝜓𝑅subscript𝑈subscript𝑓𝔭\psi_{R(U)_{f_{\mathfrak{p}}}} agrees with xa𝔭xa𝔭1maps-to𝑥subscript𝑎𝔭𝑥superscriptsubscript𝑎𝔭1x\mapsto a_{\mathfrak{p}}xa_{\mathfrak{p}}^{-1} on Mn×n(R(U)f𝔭)subscriptM𝑛𝑛𝑅subscript𝑈subscript𝑓𝔭\mathrm{M}_{n\times n}(R(U)_{f_{\mathfrak{p}}}), e.g. if they agree on an R(U)𝑅𝑈R(U)-basis of Mn×n(R(U))subscriptM𝑛𝑛𝑅𝑈\mathrm{M}_{n\times n}(R(U)). Since S=𝒪𝐘𝑆subscript𝒪𝐘S={\mathcal{O}}_{\mathbf{Y}} is a local ring object, and since S(U)=𝔭SpecS(U)S(U)f𝔭𝑆𝑈subscript𝔭Spec𝑆𝑈𝑆𝑈subscript𝑓𝔭S(U)=\sum_{{\mathfrak{p}}\in\operatorname{Spec}S(U)}S(U)f_{\mathfrak{p}}, there exists a covering {V𝔭U}𝔭subscriptsubscript𝑉𝔭𝑈𝔭\{V_{\mathfrak{p}}\to U\}_{{\mathfrak{p}}} such that f𝔭S(V𝔭)×subscript𝑓𝔭𝑆superscriptsubscript𝑉𝔭f_{\mathfrak{p}}\in{S(V_{\mathfrak{p}})^{\times}}. By construction, R(U)R(V𝔭)𝑅𝑈𝑅subscript𝑉𝔭R(U)\to R(V_{\mathfrak{p}}) factors through R(U)f𝔭𝑅subscript𝑈subscript𝑓𝔭R(U)_{f_{\mathfrak{p}}}, and thus ψ𝜓\psi is inner on V𝔭subscript𝑉𝔭V_{\mathfrak{p}} for all 𝔭𝔭{\mathfrak{p}}, as required. ∎

Lemma 4.3.10.

Let π:𝐗𝐘:𝜋𝐗𝐘\pi:{\mathbf{X}}\to{\mathbf{Y}} be a geometric morphism of topoi such that πsubscript𝜋\pi_{*} preserves epimorphisms. Then πsubscript𝜋\pi_{*} preserves quotients by equivalence relations. In particular, for any group object G𝐺G, any G𝐺G-torsor P𝑃P, and any G𝐺G-set X𝑋X, there is a canonical isomorphism π(P×GX)πP×πGπXsubscript𝜋superscript𝐺𝑃𝑋superscriptsubscript𝜋𝐺subscript𝜋𝑃subscript𝜋𝑋\pi_{*}(P\times^{G}X)\cong\pi_{*}P\times^{\pi_{*}G}\pi_{*}X.

Proof.

Recall that in a topos 𝐗𝐗{\mathbf{X}}, any equivalence relation QA×A𝑄𝐴𝐴Q\to A\times A is effective, meaning that Q=A×BA𝑄subscript𝐵𝐴𝐴Q=A\times_{B}A for some epimorphism AB𝐴𝐵A\to B—in fact, AB𝐴𝐵A\to B must be isomorphic to AA/Q𝐴𝐴𝑄A\to A/Q.

Since πsubscript𝜋\pi_{*} preserves epimorphisms and limits, this means that π(A/Q)subscript𝜋𝐴𝑄\pi_{*}(A/Q) is canonically isomorphic to πA/πQsubscript𝜋𝐴subscript𝜋𝑄\pi_{*}A/\pi_{*}Q. ∎

We now come to the main result of this section, which allows passage from Azumaya algebras in the locally ringed topos (𝐗,𝒪𝐗)𝐗subscript𝒪𝐗(\mathbf{X},\mathcal{O}_{\mathbf{X}}) to Azumaya algebras in the ringed topos (𝐘,R)𝐘𝑅(\mathbf{Y},R).

Theorem 4.3.11.

Suppose 𝐗𝐗{\mathbf{X}} is a locally ringed topos with involution λ=(Λ,ν,λ)𝜆Λ𝜈𝜆\lambda=(\Lambda,\nu,\lambda), and that π:𝐗𝐘:𝜋𝐗𝐘\pi:{\mathbf{X}}\to{\mathbf{Y}} is an exact quotient relative to λ𝜆\lambda. Let R=π𝒪𝐗𝑅subscript𝜋subscript𝒪𝐗R=\pi_{*}\mathcal{O}_{\mathbf{X}} and n𝑛n\in\mathbb{N}. Then the following categories are equivalent:

  1. (a)

    Azn(𝐗,𝒪𝐗)subscriptAz𝑛𝐗subscript𝒪𝐗{\text{\bf Az}}_{n}({\mathbf{X}},{\mathcal{O}}_{\mathbf{X}}), the category of Azumaya 𝒪𝐗subscript𝒪𝐗\mathcal{O}_{\mathbf{X}}-algebras of degree n𝑛n.

  2. (b)

    Tors(𝐗,PGLn(𝒪𝐗))Tors𝐗subscriptPGL𝑛subscript𝒪𝐗\text{\bf Tors}({\mathbf{X}},\operatorname{PGL}_{n}({\mathcal{O}}_{\mathbf{X}})), the category of PGLn(𝒪𝐗)subscriptPGL𝑛subscript𝒪𝐗\operatorname{PGL}_{n}({\mathcal{O}}_{\mathbf{X}})-torsors on 𝐗𝐗\mathbf{X}.

  3. (c)

    Azn(𝐘,R)subscriptAz𝑛𝐘𝑅{\text{\bf Az}}_{n}({\mathbf{Y}},R), the category of Azumaya R𝑅R-algebras of degree n𝑛n.

  4. (d)

    Tors(𝐘,PGLn(R))Tors𝐘subscriptPGL𝑛𝑅\text{\bf Tors}({\mathbf{Y}},\operatorname{PGL}_{n}(R)), the category of PGLn(R)subscriptPGL𝑛𝑅\operatorname{PGL}_{n}(R)-torsors on 𝐘𝐘\mathbf{Y}.

The equivalence between (a) and (b), resp. (c) and (d), is the one given in Proposition 2.5.2, and the equivalence between (a) and (c), resp. (b) and (d), is given by applying πsubscript𝜋\pi_{*}.

In the context of Example 4.3.3, where our exact quotient is induced from a good C2subscript𝐶2C_{2}-quotient of schemes π:XY:𝜋𝑋𝑌\pi:X\to Y, the theorem says that every Azumaya algebra A𝐴A over X𝑋X admits an étale covering U𝑈U of Y𝑌Y (not of X𝑋X) such that A𝐴A becomes a matrix algebra after base change to X×YUsubscript𝑌𝑋𝑈X\times_{Y}U, and every automorphism ψ𝜓\psi of A𝐴A admits an étale covering V𝑉V of Y𝑌Y (again, not of X𝑋X) such that ψ𝜓\psi becomes an inner automorphism after passing to X×YVsubscript𝑌𝑋𝑉X\times_{Y}V. Theorem 4.3.11 can be regarded as a generalization of this fact.

Proof.

The equivalence between (a) and (b), resp. (c) and (d), is Proposition 2.5.2; here we identified PGLn(R)subscriptPGL𝑛𝑅\operatorname{PGL}_{n}(R) with 𝒜utR(Mn×n(R))𝒜𝑢subscript𝑡𝑅subscriptM𝑛𝑛𝑅\mathcal{A}ut_{R}(\mathrm{M}_{n\times n}(R)) as in Lemma 4.3.9. The equivalence between (b) and (d) is Theorem 4.3.6(i), together with the isomorphism πPGLn(𝒪𝐗)PGLn(R)subscript𝜋subscriptPGL𝑛subscript𝒪𝐗subscriptPGL𝑛𝑅\pi_{*}\operatorname{PGL}_{n}({\mathcal{O}}_{{\mathbf{X}}})\cong\operatorname{PGL}_{n}(R) of Lemma 4.3.8. It now follows from Lemma 4.3.10 that the induced equivalence between (a) and (c) is given by applying πsubscript𝜋\pi_{*}. ∎

Remark 4.3.12.

The exact quotient π:𝐗𝐘:𝜋𝐗𝐘\pi:{\mathbf{X}}\to{\mathbf{Y}} gives rise to a morphism of ringed topoi with involution π^:(𝐗,𝒪𝐗)(𝐘,R):^𝜋𝐗subscript𝒪𝐗𝐘𝑅\hat{\pi}:({\mathbf{X}},{\mathcal{O}}_{\mathbf{X}})\to({\mathbf{Y}},R) by setting π^#:Rπ𝒪𝐗:subscript^𝜋#𝑅subscript𝜋subscript𝒪𝐗\hat{\pi}_{\#}:R\to\pi_{*}{\mathcal{O}}_{\mathbf{X}} to be the identity. The induced transfer functor Azn(R)Azn(𝒪𝐗)subscriptAz𝑛𝑅subscriptAz𝑛subscript𝒪𝐗{\text{\bf Az}}_{n}(R)\to{\text{\bf Az}}_{n}({\mathcal{O}}_{\mathbf{X}}) is then an inverse to the equivalence π:Azn(𝒪𝐗)Azn(R):subscript𝜋subscriptAz𝑛subscript𝒪𝐗subscriptAz𝑛𝑅\pi_{*}:{\text{\bf Az}}_{n}({\mathcal{O}}_{\mathbf{X}})\to{\text{\bf Az}}_{n}(R).

Remark 4.3.13.

As phrased, Theorem 4.3.11 addresses Azumaya 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}}-algebras of constant degree n𝑛n only. These constitute all Azumaya algebras when 𝐗𝐗{\mathbf{X}} is connected, but not in general. If we replace n𝑛n with a global section of the constant sheaf \mathbb{N} on 𝐗𝐗{\mathbf{X}}, then Theorem 4.3.11 still holds, provided that n𝑛n is fixed by Λ:H0(𝐗,)H0(𝐗,Λ)=H0(𝐗,):ΛsuperscriptH0𝐗superscriptH0𝐗ΛsuperscriptH0𝐗\Lambda:\mathrm{H}^{0}({\mathbf{X}},\mathbb{N})\to\mathrm{H}^{0}({\mathbf{X}},\Lambda\mathbb{N})=\mathrm{H}^{0}({\mathbf{X}},\mathbb{N}), in which case n𝑛n can be understood as an element of H0(𝐘,)superscriptH0𝐘\mathrm{H}^{0}({\mathbf{Y}},\mathbb{N}). Since Theorem 4.3.11 is used throughout, we tacitly assume henceforth that all Azumaya 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}}-algebras have degrees that are fixed under ΛΛ\Lambda. This makes little difference in practice, because any Azumaya 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}}-algebra is Brauer equivalent to another Azumaya 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}}-algebra of degree which is fixed by ΛΛ\Lambda.

We define an involution on the ringed topos (𝐘,R)𝐘𝑅({\mathbf{Y}},R) by setting Λ=id𝐘Λsubscriptid𝐘\Lambda=\mathrm{id}_{\mathbf{Y}}, ν=id𝜈id\nu=\mathrm{id} and λ(𝐘,R)=πλ(𝐗,𝒪𝐗)subscript𝜆𝐘𝑅subscript𝜋subscript𝜆𝐗subscript𝒪𝐗\lambda_{({\mathbf{Y}},R)}=\pi_{*}\lambda_{({\mathbf{X}},{\mathcal{O}}_{\mathbf{X}})}. Since πsubscript𝜋\pi_{*} preserves products, for any 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}}-algebra A𝐴A, we have πΛAop=ΛπAopsubscript𝜋Λsuperscript𝐴opΛsubscript𝜋superscript𝐴op\pi_{*}\Lambda A^{\text{op}}=\Lambda\pi_{*}A^{\text{op}} as R𝑅R-algebras. However, we alert the reader that while ΛπAop=πAopΛsubscript𝜋superscript𝐴opsubscript𝜋superscript𝐴op\Lambda\pi_{*}A^{\text{op}}=\pi_{*}A^{\text{op}} as noncommutative rings, the R𝑅R-algebra structure of ΛπAopΛsubscript𝜋superscript𝐴op\Lambda\pi_{*}A^{\text{op}} is obtained from the R𝑅R-algebra structure of πAopsubscript𝜋superscript𝐴op\pi_{*}A^{\text{op}} by twisting via λ:RΛR=R:𝜆𝑅Λ𝑅𝑅\lambda:R\to\Lambda R=R, as explained in Subsection 4.1. Theorem 4.3.11 now implies:

Corollary 4.3.14.

For all n𝑛n\in\mathbb{N}, the functor πsubscript𝜋\pi_{*} induces an equivalence between Azn(𝒪𝐗,λ)subscriptAz𝑛subscript𝒪𝐗𝜆{\text{\bf Az}}_{n}({\mathcal{O}}_{\mathbf{X}},\lambda), the category of degree-n𝑛n Azumaya 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}}-algebras with λ𝜆\lambda-involution, and Azn(R,λ)subscriptAz𝑛𝑅𝜆{\text{\bf Az}}_{n}(R,\lambda), the category of degree-n𝑛n Azumaya R𝑅R-algebras with λ𝜆\lambda-involution.

At this point we conclude that results proved so far allow one to shift freely between (𝐗,𝒪𝐗)𝐗subscript𝒪𝐗({\mathbf{X}},{\mathcal{O}}_{\mathbf{X}}) and (𝐘,R)𝐘𝑅({\mathbf{Y}},R), at least when Azumaya algebras, possibly with a λ𝜆\lambda-involution, are concerned. Of these two contexts, we shall work most often in the second, since this is technically easier. That said, the starting point is always a locally ringed topos (𝐗,𝒪𝐗)𝐗subscript𝒪𝐗({\mathbf{X}},{\mathcal{O}}_{\mathbf{X}}) with involution λ=(Λ,ν,λ)𝜆Λ𝜈𝜆\lambda=(\Lambda,\nu,\lambda), and the choice of the corresponding 𝐘𝐘{\mathbf{Y}}, R𝑅R and λ(𝐘,R)subscript𝜆𝐘𝑅\lambda_{({\mathbf{Y}},R)} is not in general uniquely determined by the initial data.

4.4. Examples of Exact Quotients

We now turn to providing various examples of exact C2subscript𝐶2C_{2}-quotients. In particular, we will prove that Examples 4.3.3 and 4.3.4 are exact C2subscript𝐶2C_{2}-quotients. It will also be shown that any locally ringed topos with involution admits an exact quotient.

Of the two conditions of Definition 4.3.1, condition (E2) is harder to establish. The following lemma is our main tool in proving it.

Lemma 4.4.1.

Let 𝐗𝐗\mathbf{X}, 𝐘𝐘\mathbf{Y} be topoi and let π:𝐗𝐘:𝜋𝐗𝐘\pi:\mathbf{X}\to\mathbf{Y} be a geometric morphism. Suppose that 𝐘𝐘\mathbf{Y} has a conservative family of points {pi:𝐩𝐭𝐘}iIsubscriptconditional-setsubscript𝑝𝑖𝐩𝐭𝐘𝑖𝐼\{p_{i}:\mathbf{pt}\to{\mathbf{Y}}\}_{i\in I} with the property that for each iI𝑖𝐼i\in I, there exists a set of points {jn:𝐩𝐭𝐗}nNisubscriptconditional-setsubscript𝑗𝑛𝐩𝐭𝐗𝑛subscript𝑁𝑖\{j_{n}:\mathbf{pt}\to{\mathbf{X}}\}_{n\in{N_{i}}} such that the functors UpiπUmaps-to𝑈superscriptsubscript𝑝𝑖subscript𝜋𝑈U\mapsto p_{i}^{*}\pi_{*}U and UnNijnUmaps-to𝑈subscriptproduct𝑛subscript𝑁𝑖superscriptsubscript𝑗𝑛𝑈U\mapsto\prod_{n\in N_{i}}j_{n}^{*}U from 𝐗𝐗{\mathbf{X}} to 𝐩𝐭𝐩𝐭\mathbf{pt} are isomorphic. Then πsubscript𝜋\pi_{*} preserves epimorphisms.

Proof.

If pi:𝐩𝐭𝐘:subscript𝑝𝑖𝐩𝐭𝐘p_{i}:\mathbf{pt}\to{\mathbf{Y}} is a point as in the lemma, then piπsuperscriptsubscript𝑝𝑖subscript𝜋p_{i}^{*}\pi_{*} preserves epimorphisms because each jnsuperscriptsubscript𝑗𝑛j_{n}^{*} does. By assumption, a morphism ψ𝜓\psi in 𝐘𝐘{\mathbf{Y}} is an epimorphism if and only if piψsuperscriptsubscript𝑝𝑖𝜓p_{i}^{*}\psi is an epimorphism for any pi:𝐩𝐭𝐘:subscript𝑝𝑖𝐩𝐭𝐘p_{i}:\mathbf{pt}\to{\mathbf{Y}} as in the statement, so πsubscript𝜋\pi_{*} preserves epimorphisms. ∎

Informally, a geometric morphism satisfying the conditions of Lemma 4.4.1 can be regarded as having “discrete fibres”. It can also be thought of as a generalization of a finite morphism in algebraic geometry, thanks to the following corollary.

Corollary 4.4.2.

Let π:XY:𝜋𝑋𝑌\pi:X\to Y be a finite morphism of schemes. Then the direct image functor π:Sh(Xét)Sh(Yét):subscript𝜋Shsubscript𝑋étShsubscript𝑌ét\pi_{*}:\text{\bf Sh}(X_{\text{\'{e}t}})\to\text{\bf Sh}(Y_{\text{\'{e}t}}) preserves epimorphisms.

This arises in the proof that the higher direct images vanish for cohomology with abelian coefficients, [de_jong_stacks_2017, Tag 03QN]. We have included a proof here in order to present a modification later.

Proof.

Recall ([artin_theorie_1972, Exp. VIII, §3–4]) that the points of Sh(Yét)Shsubscript𝑌ét\text{\bf Sh}(Y_{\text{\'{e}t}}) are constructed as follows: Given yY𝑦𝑌y\in Y, choose a cofiltered system of étale neighbourhoods {(Uα,uα)(Y,y)}αsubscriptsubscript𝑈𝛼subscript𝑢𝛼𝑌𝑦𝛼\{(U_{\alpha},u_{\alpha})\to(Y,y)\}_{\alpha} such that limαUα=SpecBsubscript𝛼subscript𝑈𝛼Spec𝐵\lim_{\alpha}U_{\alpha}=\operatorname{Spec}B, where B𝐵B is a strictly henselian ring, necessarily isomorphic to the strict henselization of 𝒪Y,ysubscript𝒪𝑌𝑦{\mathcal{O}}_{Y,y}. Then the functor i:FcolimαF(Uα):superscript𝑖maps-to𝐹subscriptcolim𝛼𝐹subscript𝑈𝛼i^{*}:F\mapsto\operatornamewithlimits{colim}_{\alpha}F(U_{\alpha}) from Sh(Yét)Shsubscript𝑌ét\text{\bf Sh}(Y_{\text{\'{e}t}}) to 𝐩𝐭𝐩𝐭\mathbf{pt} and its right adjoint isubscript𝑖i_{*} define a point i:𝐩𝐭Sh(Yét):𝑖𝐩𝐭Shsubscript𝑌éti:\mathbf{pt}\to\text{\bf Sh}(Y_{\text{\'{e}t}}), and these points form a conservative family, [artin_theorie_1972, Thm. VIII.3.5].

Let i:𝐩𝐭Sh(Yét):𝑖𝐩𝐭Shsubscript𝑌éti:\mathbf{pt}\to\text{\bf Sh}(Y_{\text{\'{e}t}}) be such a point and write Vα=Uα×YXsubscript𝑉𝛼subscript𝑌subscript𝑈𝛼𝑋V_{\alpha}=U_{\alpha}\times_{Y}X. For all sheaves F𝐹F on Xétsubscript𝑋étX_{\text{\'{e}t}}, we have iπF=colimαF(Vα)superscript𝑖subscript𝜋𝐹subscriptcolim𝛼𝐹subscript𝑉𝛼i^{*}\pi_{*}F=\operatornamewithlimits{colim}_{\alpha}F(V_{\alpha}). Note that limαVα=limUα×YX=SpecB×YXsubscript𝛼subscript𝑉𝛼subscript𝑌subscript𝑈𝛼𝑋Specsubscript𝑌𝐵𝑋\lim_{\alpha}V_{\alpha}=\lim U_{\alpha}\times_{Y}X=\operatorname{Spec}B\times_{Y}X. Since π𝜋\pi is finite, SpecB×YX=SpecASpecsubscript𝑌𝐵𝑋Spec𝐴\operatorname{Spec}B\times_{Y}X=\operatorname{Spec}A where A𝐴A is a finite B𝐵B-algebra.

By [de_jong_stacks_2017, Tag 04GH], A=n=1tAn𝐴superscriptsubscriptproduct𝑛1𝑡subscript𝐴𝑛A=\prod_{n=1}^{t}A_{n} where each Ansubscript𝐴𝑛A_{n} is a henselian ring. Since the residue field of each Ansubscript𝐴𝑛A_{n} is finite over the separably closed residue field of B𝐵B, each Ansubscript𝐴𝑛A_{n} is strictly henselian. Letting e1,,etsubscript𝑒1subscript𝑒𝑡e_{1},\dots,e_{t} be the primitive idempotents of A𝐴A, we may assume, by appropriately thinning the family {UαY}αsubscriptsubscript𝑈𝛼𝑌𝛼\{U_{\alpha}\to Y\}_{\alpha}, that e1,,etsubscript𝑒1subscript𝑒𝑡e_{1},\dots,e_{t} are defined as compatible global sections on each Vαsubscript𝑉𝛼V_{\alpha}. This allows us to write Vαsubscript𝑉𝛼V_{\alpha} as nVn,αsubscriptsquare-union𝑛subscript𝑉𝑛𝛼\bigsqcup_{n}V_{n,\alpha} such that limαVn,α=SpecAnsubscript𝛼subscript𝑉𝑛𝛼Specsubscript𝐴𝑛\lim_{\alpha}V_{n,\alpha}=\operatorname{Spec}A_{n} for all n𝑛n. Let xnsubscript𝑥𝑛x_{n} and vn,αsubscript𝑣𝑛𝛼v_{n,\alpha} denote the images of the closed point of SpecAnSpecsubscript𝐴𝑛\operatorname{Spec}A_{n} in X𝑋X and Vn,αsubscript𝑉𝑛𝛼V_{n,\alpha}, respectively, and let jn:𝐩𝐭Sh(Xét):subscript𝑗𝑛𝐩𝐭Shsubscript𝑋étj_{n}:\mathbf{pt}\to\text{\bf Sh}(X_{\text{\'{e}t}}) denote the point corresponding to the filtered system {(Vn,α,vn,α)(X,xn)}αsubscriptsubscript𝑉𝑛𝛼subscript𝑣𝑛𝛼𝑋subscript𝑥𝑛𝛼\{(V_{n,\alpha},v_{n,\alpha})\to(X,x_{n})\}_{\alpha}. Since we can commute a directed colimit past a finite limit, we have shown that

iπF=colimαF(Vα)=colimαnF(Vn,α)=njnF,superscript𝑖subscript𝜋𝐹subscriptcolim𝛼𝐹subscript𝑉𝛼subscriptcolim𝛼subscriptproduct𝑛𝐹subscript𝑉𝑛𝛼subscriptproduct𝑛superscriptsubscript𝑗𝑛𝐹i^{*}\pi_{*}F=\operatornamewithlimits{colim}_{\alpha}F(V_{\alpha})=\operatornamewithlimits{colim}_{\alpha}\prod_{n}F(V_{n,\alpha})=\prod_{n}j_{n}^{*}F\ ,

and the result now follows from Lemma 4.4.1. ∎

Corollary 4.4.3.

Let π:XY:𝜋𝑋𝑌\pi:X\to Y be a continuous morphism of topological spaces such that:

  1. (1)

    For any yY𝑦𝑌y\in Y and any open neighbourhood Uπ1(y)superscript𝜋1𝑦𝑈U\supseteq\pi^{-1}(y), there exists an open neighbourhood V𝑉V of y𝑦y such that π1(V)Usuperscript𝜋1𝑉𝑈\pi^{-1}(V)\subseteq U.

  2. (2)

    For any yY𝑦𝑌y\in Y, the fibre π1(y)superscript𝜋1𝑦\pi^{-1}(y) is finite and, letting x1,,xtXsubscript𝑥1subscript𝑥𝑡𝑋x_{1},\dots,x_{t}\in X denote the points lying over y𝑦y, there exist disjoint open sets {Ui}i=1tsuperscriptsubscriptsubscript𝑈𝑖𝑖1𝑡\{U_{i}\}_{i=1}^{t} such that xiUisubscript𝑥𝑖subscript𝑈𝑖x_{i}\in U_{i}.

Then π:Sh(X)Sh(Y):subscript𝜋Sh𝑋Sh𝑌\pi_{*}:\text{\bf Sh}(X)\to\text{\bf Sh}(Y) preserves epimorphisms.

The hypotheses are satisfied when π:YX:𝜋𝑌𝑋\pi:Y\to X is a finite covering space map of Hausdorff spaces, or a closed embedding of Hausdorff spaces, for instance.

It is also easy to see that condition (1) is equivalent to π𝜋\pi being closed, and condition (2) is equivalent to π𝜋\pi having finite fibres and being separated in the sense that the image of the diagonal map XX×YX𝑋subscript𝑌𝑋𝑋X\to X\times_{Y}X is closed.

Proof.

Again, we use Lemma 4.4.1. For Sh(Y)Sh𝑌\text{\bf Sh}(Y), the points are induced by inclusion maps i:{y}Y:𝑖𝑦𝑌i:\{y\}\to Y as y𝑦y rages over Y𝑌Y, [mac_lane_sheaves_1992, Chap. VII, §5]. Fix such an inclusion, let x1,,xtsubscript𝑥1subscript𝑥𝑡x_{1},\dots,x_{t} denote the points in π1(y)superscript𝜋1𝑦\pi^{-1}(y), and let jn:{xn}X:subscript𝑗𝑛subscript𝑥𝑛𝑋j_{n}:\{x_{n}\}\to X denote the inclusion maps. The corresponding morphisms on the topoi of sheaves will be denoted by the same letters.

By definition, for any sheaf F𝐹F on X𝑋X, we have

iπ=colimUy(π1(U))superscript𝑖subscript𝜋subscriptcolim𝑦𝑈superscript𝜋1𝑈i^{*}\pi_{*}\mathcal{F}=\operatornamewithlimits{colim}_{U\ni y}\mathcal{F}(\pi^{-1}(U))

where the colimit is taken over all open neighbourhoods of y𝑦y. Using condition (2), choose disjoint open neighbourhoods {Vn}n=1tsuperscriptsubscriptsubscript𝑉𝑛𝑛1𝑡\{V_{n}\}_{n=1}^{t} with xnVnsubscript𝑥𝑛subscript𝑉𝑛x_{n}\in V_{n}. Condition (1) implies that the family {Vnπ1(U)|U is an open neighbourhood of y}conditional-setsubscript𝑉𝑛superscript𝜋1𝑈U is an open neighbourhood of y\{{V_{n}\cap\pi^{-1}(U)}\,|\,\text{$U$ is an open neighbourhood of $y$}\} is a basis of open neighbourhoods of xnsubscript𝑥𝑛x_{n}, hence

colimUy(π1(U))=n=1tcolimUy(π1(U)Vn)=n=1tjn.subscriptcolim𝑦𝑈superscript𝜋1𝑈superscriptsubscriptproduct𝑛1𝑡subscriptcolim𝑦𝑈superscript𝜋1𝑈subscript𝑉𝑛superscriptsubscriptproduct𝑛1𝑡superscriptsubscript𝑗𝑛\operatornamewithlimits{colim}_{U\ni y}\mathcal{F}(\pi^{-1}(U))=\prod_{n=1}^{t}\operatornamewithlimits{colim}_{U\ni y}\mathcal{F}(\pi^{-1}(U)\cap V_{n})=\prod_{n=1}^{t}j_{n}^{*}\mathcal{F}\ .

It follows that iπ=n=1tjnsuperscript𝑖subscript𝜋superscriptsubscriptproduct𝑛1𝑡superscriptsubscript𝑗𝑛i^{*}\pi_{*}=\prod_{n=1}^{t}j_{n}^{*}, so the proof is complete. ∎

Theorem 4.4.4.

Suppose that

  1. (i)

    X𝑋X is a scheme, λ:XX:𝜆𝑋𝑋\lambda:X\to X is an involution, π:XY:𝜋𝑋𝑌\pi:X\to Y is a good quotient relative to {1,λ}1𝜆\{1,\lambda\}, (𝐗,𝒪𝐗)=(Sh(Xét),𝒪X)𝐗subscript𝒪𝐗Shsubscript𝑋étsubscript𝒪𝑋({\mathbf{X}},{\mathcal{O}}_{\mathbf{X}})=(\text{\bf Sh}(X_{\text{\'{e}t}}),{\mathcal{O}}_{X}) and (𝐘,𝒪𝐘)=(Sh(Yét),𝒪Y)𝐘subscript𝒪𝐘Shsubscript𝑌étsubscript𝒪𝑌({\mathbf{Y}},{\mathcal{O}}_{\mathbf{Y}})=(\text{\bf Sh}(Y_{\text{\'{e}t}}),{\mathcal{O}}_{Y}) (Example 4.3.3), or

  2. (ii)

    X𝑋X is a Hausdorff topological space, λ:XX:𝜆𝑋𝑋\lambda:X\to X is an involution, Y=X/{1,λ}𝑌𝑋1𝜆Y=X/\{1,\lambda\} and π:XY:𝜋𝑋𝑌\pi:X\to Y is the quotient map, (𝐗,𝒪𝐗)=(Sh(X),𝒞(X,))𝐗subscript𝒪𝐗Sh𝑋𝒞𝑋({\mathbf{X}},{\mathcal{O}}_{\mathbf{X}})=(\text{\bf Sh}(X),{\mathcal{C}}(X,\mathbb{C})) and (𝐘,𝒪𝐘)=(Sh(Y),𝒞(Y,))𝐘subscript𝒪𝐘Sh𝑌𝒞𝑌({\mathbf{Y}},{\mathcal{O}}_{\mathbf{Y}})=(\text{\bf Sh}(Y),{\mathcal{C}}(Y,\mathbb{C})) (Example 4.3.4).

Then the morphism π:(𝐗,𝒪𝐗)(𝐘,𝒪𝐘):𝜋𝐗subscript𝒪𝐗𝐘subscript𝒪𝐘\pi:(\mathbf{X},{\mathcal{O}}_{{\mathbf{X}}})\to(\mathbf{Y},{\mathcal{O}}_{{\mathbf{Y}}}) induced by π:XY:𝜋𝑋𝑌\pi:X\to Y is an exact quotient relative to the involution induced by λ𝜆\lambda.

Proof.
  1. (i)

    The fact that π:Sh(Xét)Sh(Yét):subscript𝜋Shsubscript𝑋étShsubscript𝑌ét\pi_{*}:\text{\bf Sh}(X_{\text{\'{e}t}})\to\text{\bf Sh}(Y_{\text{\'{e}t}}) preserves epimorphisms is shown as in the proof of Corollary 4.4.2, except one has to replace [de_jong_stacks_2017, Tag 04GH] by Corollary 3.3.11. Checking that 𝒪𝐘subscript𝒪𝐘{\mathcal{O}}_{\mathbf{Y}} is the coequalizer of λ,id:π𝒪𝐗π𝒪𝐗:𝜆idsubscript𝜋subscript𝒪𝐗subscript𝜋subscript𝒪𝐗\lambda,\mathrm{id}:\pi_{*}{\mathcal{O}}_{{\mathbf{X}}}\to\pi_{*}{\mathcal{O}}_{\mathbf{X}}, amounts to showing that for any étale morphism UY𝑈𝑌U\to Y, H0(U,𝒪U)superscriptH0𝑈subscript𝒪𝑈\mathrm{H}^{0}(U,{\mathcal{O}}_{U}) is the fixed ring of λ𝜆\lambda in H0(U×YX,𝒪U×YX)superscriptH0subscript𝑌𝑈𝑋subscript𝒪subscript𝑌𝑈𝑋\mathrm{H}^{0}(U\times_{Y}X,{\mathcal{O}}_{U\times_{Y}X}). In fact, it is enough to check this after base changing to an open affine covering {YiY}subscript𝑌𝑖𝑌\{Y_{i}\to Y\}, so we may assume that UY𝑈𝑌U\to Y factors as UY0Y𝑈subscript𝑌0𝑌U\to Y_{0}\to Y with Y0subscript𝑌0Y_{0} open and affine. Write Y0=SpecBsubscript𝑌0Spec𝐵Y_{0}=\operatorname{Spec}B and U=SpecB𝑈Specsuperscript𝐵U=\operatorname{Spec}B^{\prime}. Since XY𝑋𝑌X\to Y is affine, we may further write Y0×YX=SpecAsubscript𝑌subscript𝑌0𝑋Spec𝐴Y_{0}\times_{Y}X=\operatorname{Spec}A. The assumption that XY𝑋𝑌X\to Y is a good quotient relative to {1,λ}1𝜆\{1,\lambda\} implies that the sequence of B𝐵B-modules 0BAaaaλA0𝐵𝐴maps-to𝑎𝑎superscript𝑎𝜆𝐴0\to B\to A\xrightarrow{a\mapsto a-a^{\lambda}}A is exact. Since Bsuperscript𝐵B^{\prime} is flat over B𝐵B, the sequence 0BABBaaaλABB0superscript𝐵subscripttensor-product𝐵𝐴superscript𝐵maps-to𝑎𝑎superscript𝑎𝜆subscripttensor-product𝐵𝐴superscript𝐵0\to B^{\prime}\to{A\otimes_{B}B^{\prime}}\xrightarrow{a\mapsto a-a^{\lambda}}A\otimes_{B}B^{\prime} is exact, and hence B=H0(U,𝒪U)superscript𝐵superscriptH0𝑈subscript𝒪𝑈B^{\prime}=\mathrm{H}^{0}(U,{\mathcal{O}}_{U}) is the fixed ring of λ𝜆\lambda in ABB=H0(U×YX,𝒪U×YX)subscripttensor-product𝐵𝐴superscript𝐵superscriptH0subscript𝑌𝑈𝑋subscript𝒪subscript𝑌𝑈𝑋A\otimes_{B}B^{\prime}=\mathrm{H}^{0}(U\times_{Y}X,{\mathcal{O}}_{U\times_{Y}X}).

  2. (ii)

    Conditions (1) and (2) of Corollary 4.4.3 are easily seen to hold, hence πsubscript𝜋\pi_{*} preserves epimorphisms. It remains to show that 𝒪Ysubscript𝒪𝑌\mathcal{O}_{Y} is the equalizer of π𝒪Xsubscript𝜋subscript𝒪𝑋\pi_{*}\mathcal{O}_{X} under the action of C2={1,λ}subscript𝐶21𝜆C_{2}=\{1,\lambda\}. To this end, let UY𝑈𝑌U\subseteq Y be an open set. The C2subscript𝐶2C_{2}-action on X𝑋X restricts to an action on π1(U)superscript𝜋1𝑈\pi^{-1}(U) and π1(U)/C2=Usuperscript𝜋1𝑈subscript𝐶2𝑈\pi^{-1}(U)/C_{2}=U. In particular, 𝒪Y(U)=C(U,)subscript𝒪𝑌𝑈𝐶𝑈\mathcal{O}_{Y}(U)=C(U,\mathbb{C}) is in natural bijection with the set of functions in π𝒪X(U)=𝒪X(π1(U))=C(π1(U),)subscript𝜋subscript𝒪𝑋𝑈subscript𝒪𝑋superscript𝜋1𝑈𝐶superscript𝜋1𝑈\pi_{*}\mathcal{O}_{X}(U)=\mathcal{O}_{X}(\pi^{-1}(U))=C(\pi^{-1}(U),\mathbb{C}) that are fixed under the C2subscript𝐶2C_{2}-action. This means that 𝒪Ysubscript𝒪𝑌\mathcal{O}_{Y} is the fixed subsheaf of π𝒪Xsubscript𝜋subscript𝒪𝑋\pi_{*}\mathcal{O}_{X} under the action of C2subscript𝐶2C_{2}.∎

Remark 4.4.5.

The proofs of Theorem 4.4.4(i) and Corollary 4.4.2 can be modified to work for the Nisnevich site of a scheme — simply replace étale neighbourhoods by Nisnevich neighbourhoods and strictly henselian rings by henselian rings. Disregarding set-theoretic problems, the large étale and Nisnevich sites can be handled similarly, using suitable conservative families of points, provided one assumes in Theorem 4.4.4(i) that 𝒪Yπ𝒪Xλidπ𝒪Xsubscript𝒪𝑌subscript𝜋subscript𝒪𝑋𝜆idsubscript𝜋subscript𝒪𝑋{\mathcal{O}}_{Y}\to\pi_{*}{\mathcal{O}}_{X}\xrightarrow{\lambda-\mathrm{id}}\pi_{*}{\mathcal{O}}_{X} splits in the middle, which is the case when 2𝒪X×2superscriptsubscript𝒪𝑋2\in{{\mathcal{O}}_{X}^{\times}}. This assumption guarantees that the sequence remains exact after base-change to any Y𝑌Y-scheme U𝑈U, not necessarily flat.

Likewise, the proofs Theorem 4.4.4(ii) and Corollary 4.4.3 can be modified to work for the large site of a topological space.

The next examples bring several situations where condition (E1) of Definition 4.3.1 is satisfied while condition (E2) is not. They also show that some of the assumptions made in Theorem 4.4.4 cannot be removed in general.

Example 4.4.6.

Let R𝑅R be a strictly henselian discrete valuation ring with fraction field K𝐾K. Let X𝑋X denote the scheme obtained by gluing two copies of Y:=SpecRassign𝑌Spec𝑅Y:=\operatorname{Spec}R along SpecKSpec𝐾\operatorname{Spec}K, and let λ:XX:𝜆𝑋𝑋\lambda:X\to X denote the involution exchanging these two copies. The morphism π:XY:𝜋𝑋𝑌\pi:X\to Y which restricts to the identity on each of the copies of SpecRSpec𝑅\operatorname{Spec}R is a geometric quotient relative to C2={1,λ}subscript𝐶21𝜆C_{2}=\{1,\lambda\} in the sense of [de_jong_stacks_2017, Tag 04AD], namely, 𝒪Y=(π𝒪X)C2subscript𝒪𝑌superscriptsubscript𝜋subscript𝒪𝑋subscript𝐶2{\mathcal{O}}_{Y}=(\pi_{*}{\mathcal{O}}_{X})^{C_{2}}, Y=X/C2𝑌𝑋subscript𝐶2Y=X/C_{2} as topological spaces, and the latter property holds after base change. In particular, π:XY:𝜋𝑋𝑌\pi:X\to Y is the C2subscript𝐶2C_{2}-quotient of X𝑋X in the category of schemes. However, the induced C2subscript𝐶2C_{2}-equivariant morphism π:(Sh(Xét),𝒪X)(Sh(Yét),𝒪Y):𝜋Shsubscript𝑋étsubscript𝒪𝑋Shsubscript𝑌étsubscript𝒪𝑌\pi:(\text{\bf Sh}(X_{\text{\'{e}t}}),{\mathcal{O}}_{X})\to(\text{\bf Sh}(Y_{\text{\'{e}t}}),{\mathcal{O}}_{Y}) is not an exact quotient, the reason being that πsubscript𝜋\pi_{*} does not preserve epimorphisms.

To see this, fix a non-trivial abelian group A𝐴A, which will be regarded as a constant sheaf on the appropriate space, and let i,j:SpecRX:𝑖𝑗Spec𝑅𝑋i,j:\operatorname{Spec}R\to X denote the inclusions of the two copies of SpecRSpec𝑅\operatorname{Spec}R in X𝑋X. Since i𝑖i is an open immersion, we can form the extension-by-00 functor i!:Sh((SpecR)ét)Sh(Xét):subscript𝑖ShsubscriptSpec𝑅étShsubscript𝑋éti_{!}:\text{\bf Sh}((\operatorname{Spec}R)_{\text{\'{e}t}})\to\text{\bf Sh}(X_{\text{\'{e}t}}), which is left adjoint to isuperscript𝑖i^{*}. Let F=i!iAj!jA𝐹direct-sumsubscript𝑖superscript𝑖𝐴subscript𝑗superscript𝑗𝐴F=i_{!}i^{*}A\oplus j_{!}j^{*}A. The counit maps ε(i):i!iAA:superscript𝜀𝑖subscript𝑖superscript𝑖𝐴𝐴\varepsilon^{(i)}:i_{!}i^{*}A\to A and ε(j):j!jAA:superscript𝜀𝑗subscript𝑗superscript𝑗𝐴𝐴\varepsilon^{(j)}:j_{!}j^{*}A\to A give rise to a morphism ψ:FA:𝜓𝐹𝐴\psi:F\to A given by (xy)(ε(i)x+ε(j)y)maps-todirect-sum𝑥𝑦superscript𝜀𝑖𝑥superscript𝜀𝑗𝑦(x\oplus y)\mapsto(\varepsilon^{(i)}x+\varepsilon^{(j)}y) on sections. This morphism is surjective, as can be easily seen by checking the stalks. However πψ:πFπA:subscript𝜋𝜓subscript𝜋𝐹subscript𝜋𝐴\pi_{*}\psi:\pi_{*}F\to\pi_{*}A is not surjective, as can be seen by noting that πF(Y)=0subscript𝜋𝐹𝑌0\pi_{*}F(Y)=0, πA(Y)=A0subscript𝜋𝐴𝑌𝐴0\pi_{*}A(Y)=A\neq 0, and any étale covering of Y𝑌Y has a section, because R𝑅R is strictly henselian.

This example does not stand in contradiction to Theorem 4.4.4(i) because π:XY:𝜋𝑋𝑌\pi:X\to Y is not affine, and hence not a good quotient.

Example 4.4.7.

Let X𝑋X be an infinite set endowed with the cofinite topology, let λ:XX:𝜆𝑋𝑋\lambda:X\to X be an involution acting freely on X𝑋X, and let π:XY=X/C2:𝜋𝑋𝑌𝑋subscript𝐶2\pi:X\to Y=X/C_{2} be the quotient map. Then the induced C2subscript𝐶2C_{2}-equivariant morphism π:(Sh(X),𝒞(X,))(Sh(Y),𝒞(Y,)):𝜋Sh𝑋𝒞𝑋Sh𝑌𝒞𝑌\pi:(\text{\bf Sh}(X),{\mathcal{C}}(X,\mathbb{C}))\to(\text{\bf Sh}(Y),{\mathcal{C}}(Y,\mathbb{C})) is not an exact C2subscript𝐶2C_{2}-quotient, because πsubscript𝜋\pi_{*} fails to preserve epimorphisms. This is shown as in Example 4.4.6, except here one chooses xX𝑥𝑋x\in X and uses the open embeddings i:X{x}X:𝑖𝑋𝑥𝑋i:X-\{x\}\to X and j:X{λ(x)}X:𝑗𝑋𝜆𝑥𝑋j:X-\{\lambda(x)\}\to X. We conclude that in Theorem 4.4.4(ii), the assumption that X𝑋X is Hausdorff in cannot be removed in general, even when λ𝜆\lambda acts freely on X𝑋X.

Example 4.4.8.

Let R𝑅R be a principal ideal domain admitting exactly two maximal ideals, 𝔞𝔞{\mathfrak{a}} and 𝔟𝔟{\mathfrak{b}}. Suppose that there exists an involution λ:RR:𝜆𝑅𝑅\lambda:R\to R exchanging 𝔞𝔞{\mathfrak{a}} and 𝔟𝔟{\mathfrak{b}}, and moreover, that the fixed ring of λ𝜆\lambda, denoted S𝑆S, is a discrete valuation ring. Let X=SpecR𝑋Spec𝑅X=\operatorname{Spec}R, Y=SpecS𝑌Spec𝑆Y=\operatorname{Spec}S and let π:XY:𝜋𝑋𝑌\pi:X\to Y be the morphism adjoint to the inclusion SR𝑆𝑅S\to R. Then π:XY:𝜋𝑋𝑌\pi:X\to Y is a good quotient relative to λ:XX:𝜆𝑋𝑋\lambda:X\to X, but the induced C2subscript𝐶2C_{2}-equivariant morphism π:(Sh(XZar),𝒪X)(Sh(YZar),𝒪Y):𝜋Shsubscript𝑋Zarsubscript𝒪𝑋Shsubscript𝑌Zarsubscript𝒪𝑌\pi:(\text{\bf Sh}(X_{\operatorname{Zar}}),{\mathcal{O}}_{X})\to(\text{\bf Sh}(Y_{\operatorname{Zar}}),{\mathcal{O}}_{Y}) is not an exact quotient, because, yet again, πsubscript𝜋\pi_{*} does not preserve epimorphisms. Again, this is checked as in Example 4.4.6 by using the open embeddings i:SpecR𝔞SpecR:𝑖Specsubscript𝑅𝔞Spec𝑅i:\operatorname{Spec}R_{\mathfrak{a}}\to\operatorname{Spec}R and j:SpecR𝔟SpecR:𝑗Specsubscript𝑅𝔟Spec𝑅j:\operatorname{Spec}R_{\mathfrak{b}}\to\operatorname{Spec}R. This shows that we cannot, in general, replace the étale site with the Zariski site in Theorem 4.4.4(i), even when π:XY:𝜋𝑋𝑌\pi:X\to Y is quadratic étale.

Remark 4.4.9.

Let X𝑋X be a scheme, let λ:XX:𝜆𝑋𝑋\lambda:X\to X be an involution and let π:XY:𝜋𝑋𝑌\pi:X\to Y be a good quotient relative to {1,λ}1𝜆\{1,\lambda\}. Then the associated morphism of fppffppf\mathrm{fppf} topoi π:(Sh(Xfppf),𝒪X)(Sh(Yfppf),𝒪Y):𝜋Shsubscript𝑋fppfsubscript𝒪𝑋Shsubscript𝑌fppfsubscript𝒪𝑌\pi:(\text{\bf Sh}(X_{\mathrm{fppf}}),{\mathcal{O}}_{X})\to(\text{\bf Sh}(Y_{\mathrm{fppf}}),{\mathcal{O}}_{Y}) is not exact in general, even when π𝜋\pi is an fppf morphism.

For example, let k𝑘k be a field characteristic 2absent2\neq 2, and consider X=Speck[ε|ε2=0]𝑋Spec𝑘delimited-[]conditional𝜀superscript𝜀20X=\operatorname{Spec}k[\varepsilon\,|\,\varepsilon^{2}=0], Y=Speck𝑌Spec𝑘Y=\operatorname{Spec}k and the k𝑘k-involution λ𝜆\lambda sending ε𝜀\varepsilon to ε𝜀-\varepsilon. Then xx2:𝒪X𝒪X:maps-to𝑥superscript𝑥2subscript𝒪𝑋subscript𝒪𝑋x\mapsto x^{2}:{\mathcal{O}}_{X}\to{\mathcal{O}}_{X} is surjective as morphism in Sh(Xfppf)Shsubscript𝑋fppf\text{\bf Sh}(X_{\mathrm{fppf}}), but its pushforward to Sh(Yfppf)Shsubscript𝑌fppf\text{\bf Sh}(Y_{\mathrm{fppf}}) is not, because ε𝜀\varepsilon is not in the image of xx2:A[ε|ε2=0]A[ε|ε2=0]:maps-to𝑥superscript𝑥2𝐴delimited-[]conditional𝜀superscript𝜀20𝐴delimited-[]conditional𝜀superscript𝜀20x\mapsto x^{2}:A[\varepsilon\,|\,\varepsilon^{2}=0]\to A[\varepsilon\,|\,\varepsilon^{2}=0] for all commutative k𝑘k-algebras A𝐴A.

Nevertheless, when π:XY:𝜋𝑋𝑌\pi:X\to Y is finite and locally free, Theorem 4.3.11 and Corollary 4.3.14 still hold, the reason being that PGLn(𝒪X)subscriptPGL𝑛subscript𝒪𝑋\operatorname{PGL}_{n}({\mathcal{O}}_{X}) and PGLn(R)=πPGLn(𝒪X)subscriptPGL𝑛𝑅subscript𝜋subscriptPGL𝑛subscript𝒪𝑋\operatorname{PGL}_{n}(R)=\pi_{*}\operatorname{PGL}_{n}({\mathcal{O}}_{X}) are both represented by smooth affine group schemes over X𝑋X (use [bosch_1990_Neron_models, Prop. 7.6.5(h)]), and hence their étale and fppf cohomologies coincide [grothendieck_1968_groupe_de_Brauer_III, Thm. 11.7, Rmk. 11.8(3)]. As a result, some theorems in the next sections, e.g. Theorems 5.2.13 and 6.3.3, also hold in the context of fppffppf\mathrm{fppf} ringed topoi associated to a finite locally free good C2subscript𝐶2C_{2}-quotient of schemes π:XY:𝜋𝑋𝑌\pi:X\to Y.

We finish with demonstrating that every locally ringed topos 𝐗𝐗{\mathbf{X}} with involution λ=(Λ,ν,λ)𝜆Λ𝜈𝜆\lambda=(\Lambda,\nu,\lambda) admits a canonical exact quotient, sometimes called the “homotopy fixed points”, as in [merling_equivariant_2017, Section 2]. We denote this exact quotient by π:𝐗[𝐗/C2]:𝜋𝐗delimited-[]𝐗subscript𝐶2\pi:{\mathbf{X}}\to[{\mathbf{X}}/C_{2}].

As the notation suggests, when (𝐗,𝒪𝐗)=(Sh(Xét),𝒪X)𝐗subscript𝒪𝐗Shsubscript𝑋étsubscript𝒪𝑋({\mathbf{X}},{\mathcal{O}}_{\mathbf{X}})=(\text{\bf Sh}(X_{\text{\'{e}t}}),{\mathcal{O}}_{X}) for a scheme X𝑋X, the ringed topos [𝐗/C2]delimited-[]𝐗subscript𝐶2[{\mathbf{X}}/C_{2}] will be equivalent to the étale ringed topos of the Deligne–Mumford stack [X/C2]delimited-[]𝑋subscript𝐶2[X/C_{2}]. Indeed, the objects of [𝐗/C2]delimited-[]𝐗subscript𝐶2[{\mathbf{X}}/C_{2}] will be C2subscript𝐶2C_{2}-equivariant sheaves, the data of which are equivalent to specifying a sheaf on the étale site of [X/C2]delimited-[]𝑋subscript𝐶2[X/C_{2}]; this is explained for coherent sheaves in [vistoli_1989_intersection_theory, Example 7.21], but the principle works for set-valued sheaves (in the sense of [de_jong_stacks_2017, Tag 06TN]) as well.

Construction 4.4.10.

Define the category [𝐗/C2]delimited-[]𝐗subscript𝐶2[{\mathbf{X}}/C_{2}] as follows: The objects of [𝐗/C2]delimited-[]𝐗subscript𝐶2[{\mathbf{X}}/C_{2}] consist of pairs (U,τ)𝑈𝜏(U,\tau), where U𝑈U is an object of 𝐗𝐗{\mathbf{X}} and τ:UΛU:𝜏𝑈Λ𝑈\tau:U\to\Lambda U is a morphism satisfying Λττ=idUΛ𝜏𝜏subscriptid𝑈\Lambda\tau\circ\tau=\mathrm{id}_{U}. In other words, the objects of [𝐗/C2]delimited-[]𝐗subscript𝐶2[{\mathbf{X}}/C_{2}] are objects of 𝐗𝐗{\mathbf{X}} equipped with an involution, or a C2subscript𝐶2C_{2}-action. Morphisms in [𝐗/C2]delimited-[]𝐗subscript𝐶2[{\mathbf{X}}/C_{2}] are defined as commuting squares

U𝑈\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}τ𝜏\scriptstyle{\tau}f𝑓\scriptstyle{f}ΛUΛ𝑈\textstyle{\Lambda U\ignorespaces\ignorespaces\ignorespaces\ignorespaces}ΛfΛ𝑓\scriptstyle{\Lambda f}Usuperscript𝑈\textstyle{U^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}τsuperscript𝜏\scriptstyle{\tau^{\prime}}ΛU.Λsuperscript𝑈\textstyle{\Lambda U^{\prime}.}

Define π:[𝐗/C2]𝐗:superscript𝜋delimited-[]𝐗subscript𝐶2𝐗\pi^{*}:[{\mathbf{X}}/C_{2}]\to{\mathbf{X}} to be the forgetful functor (U,τ)Umaps-to𝑈𝜏𝑈(U,\tau)\mapsto U, and define π:𝐗[𝐗/C2]:subscript𝜋𝐗delimited-[]𝐗subscript𝐶2\pi_{*}:{\mathbf{X}}\to[{\mathbf{X}}/C_{2}] to be the functor sending U𝑈U to (U×ΛU,τU)𝑈Λ𝑈subscript𝜏𝑈(U\times\Lambda U,\tau_{U}) where τU:U×ΛUΛ(U×ΛU)=ΛU×U:subscript𝜏𝑈𝑈Λ𝑈Λ𝑈Λ𝑈Λ𝑈𝑈\tau_{U}:U\times\Lambda U\to\Lambda(U\times\Lambda U)=\Lambda U\times U is the interchange morphism. For a morphism ϕ:UV:italic-ϕ𝑈𝑉\phi:U\to V in 𝐗𝐗{\mathbf{X}}, let πϕ=ϕ×Λϕsubscript𝜋italic-ϕitalic-ϕΛitalic-ϕ\pi_{*}\phi=\phi\times\Lambda\phi. The functor πsuperscript𝜋\pi^{*} is easily seen to be left adjoint to πsubscript𝜋\pi_{*} with the unit and counit of the adjunction given by u(u,τu):(U,τ)ππ(U,τ)=(U×ΛU,τU):maps-to𝑢𝑢𝜏𝑢𝑈𝜏subscript𝜋superscript𝜋𝑈𝜏𝑈Λ𝑈subscript𝜏𝑈u\mapsto(u,\tau u):(U,\tau)\to\pi_{*}\pi^{*}(U,\tau)=(U\times\Lambda U,\tau_{U}) and (v,v)v:ππV=V×ΛVV:maps-to𝑣superscript𝑣𝑣superscript𝜋subscript𝜋𝑉𝑉Λ𝑉𝑉(v,v^{\prime})\mapsto v:\pi^{*}\pi_{*}V=V\times\Lambda V\to V on the level of sections (in 𝐗𝐗{\mathbf{X}}).

For objects V𝑉V in 𝐗𝐗{\mathbf{X}} and (U,τ)𝑈𝜏(U,\tau) in [𝐗/C2]delimited-[]𝐗subscript𝐶2[{\mathbf{X}}/C_{2}], let α,Vsubscript𝛼𝑉\alpha_{*,V} denote the interchange morphism (ΛV×V,τΛV)(V×ΛV,τV)Λ𝑉𝑉subscript𝜏Λ𝑉𝑉Λ𝑉subscript𝜏𝑉(\Lambda V\times V,\tau_{\Lambda V})\to(V\times\Lambda V,\tau_{V}) and let α(U,τ)subscriptsuperscript𝛼𝑈𝜏\alpha^{*}_{(U,\tau)} denote τ:UΛU:𝜏𝑈Λ𝑈\tau:U\to\Lambda U. Then αsubscript𝛼\alpha_{*} is a natural isomorphism πΛπsubscript𝜋Λsubscript𝜋\pi_{*}\Lambda\Rightarrow\pi_{*} and αsuperscript𝛼\alpha^{*} is a natural isomorphism πΛπsuperscript𝜋Λsuperscript𝜋\pi^{*}\Rightarrow\Lambda\pi^{*}. We alert the reader that these natural isomorphisms are in general not the identity transformations, even when the involution λ𝜆\lambda is trivial.

Define the ring object 𝒪[𝐗/C2]subscript𝒪delimited-[]𝐗subscript𝐶2{\mathcal{O}}_{[{\mathbf{X}}/C_{2}]} in [𝐗/C2]delimited-[]𝐗subscript𝐶2[{\mathbf{X}}/C_{2}] to be (𝒪𝐗,λ)subscript𝒪𝐗𝜆({\mathcal{O}}_{\mathbf{X}},\lambda) with the obvious ring structure. Finally, define π#:𝒪[𝐗/C2]π𝒪𝐗:subscript𝜋#subscript𝒪delimited-[]𝐗subscript𝐶2subscript𝜋subscript𝒪𝐗\pi_{\#}:{\mathcal{O}}_{[{\mathbf{X}}/C_{2}]}\to\pi_{*}{\mathcal{O}}_{\mathbf{X}} to be (𝒪𝐗,λ)(𝒪𝐗×Λ𝒪𝐗,τ𝒪𝐗)subscript𝒪𝐗𝜆subscript𝒪𝐗Λsubscript𝒪𝐗subscript𝜏subscript𝒪𝐗({\mathcal{O}}_{\mathbf{X}},\lambda)\to({\mathcal{O}}_{\mathbf{X}}\times\Lambda{\mathcal{O}}_{\mathbf{X}},\tau_{{\mathcal{O}}_{\mathbf{X}}}), where the underlying morphism 𝒪𝐗𝒪𝐗×Λ𝒪𝐗subscript𝒪𝐗subscript𝒪𝐗Λsubscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}}\to{\mathcal{O}}_{\mathbf{X}}\times\Lambda{\mathcal{O}}_{\mathbf{X}} is given by x(x,xλ)maps-to𝑥𝑥superscript𝑥𝜆x\mapsto(x,x^{\lambda}) on sections.

Proposition 4.4.11.

In Construction 4.4.10, the following hold:

  1. (i)

    [𝐗/C2]delimited-[]𝐗subscript𝐶2[{\mathbf{X}}/C_{2}] is a Grothendieck topos.

  2. (ii)

    π:=(π,π):𝐗[𝐗/C2]:assign𝜋superscript𝜋subscript𝜋𝐗delimited-[]𝐗subscript𝐶2\pi:=(\pi^{*},\pi_{*}):{\mathbf{X}}\to[{\mathbf{X}}/C_{2}] is an essential geometric morphism of topoi.

  3. (iii)

    A family of morphisms {(Ui,τi)(U,τ)}iIsubscriptsubscript𝑈𝑖subscript𝜏𝑖𝑈𝜏𝑖𝐼\{(U_{i},\tau_{i})\to(U,\tau)\}_{i\in I} in [𝐗/C2]delimited-[]𝐗subscript𝐶2[{\mathbf{X}}/C_{2}] is a covering if and only if {UiU}iIsubscriptsubscript𝑈𝑖𝑈𝑖𝐼\{U_{i}\to U\}_{i\in I} is a covering in 𝐗𝐗{\mathbf{X}}.

  4. (iv)

    𝒪[𝐗/C2]subscript𝒪delimited-[]𝐗subscript𝐶2{\mathcal{O}}_{[{\mathbf{X}}/C_{2}]} is a local ring object in [𝐗/C2]delimited-[]𝐗subscript𝐶2[{\mathbf{X}}/C_{2}].

  5. (v)

    (π,π,π#,α,α)superscript𝜋subscript𝜋subscript𝜋#superscript𝛼subscript𝛼(\pi^{*},\pi_{*},\pi_{\#},\alpha^{*},\alpha_{*}) defines an exact quotient π:(𝐗,𝒪𝐗)([𝐗/C2],𝒪[𝐗/C2]):𝜋𝐗subscript𝒪𝐗delimited-[]𝐗subscript𝐶2subscript𝒪delimited-[]𝐗subscript𝐶2\pi:({\mathbf{X}},{\mathcal{O}}_{\mathbf{X}})\to([{\mathbf{X}}/C_{2}],{\mathcal{O}}_{[{\mathbf{X}}/C_{2}]}) relative to λ𝜆\lambda.

Proof.

We first introduce the functor π!:𝐗[𝐗/C2]:subscript𝜋𝐗delimited-[]𝐗subscript𝐶2\pi_{!}:{\mathbf{X}}\to[{\mathbf{X}}/C_{2}] given by sending an object U𝑈U to (UΛU,σU)square-union𝑈Λ𝑈subscript𝜎𝑈(U\sqcup\Lambda U,\sigma_{U}) where σU:UΛUΛ(UΛU)=ΛUU:subscript𝜎𝑈square-union𝑈Λ𝑈Λsquare-union𝑈Λ𝑈square-unionΛ𝑈𝑈\sigma_{U}:U\sqcup\Lambda U\to\Lambda(U\sqcup\Lambda U)=\Lambda U\sqcup U is the interchange morphism. It is routine to check that π!subscript𝜋\pi_{!} is left adjoint to πsuperscript𝜋\pi^{*}; the unit map is the canonical embedding Vππ!V=VΛV𝑉superscript𝜋subscript𝜋𝑉square-union𝑉Λ𝑉V\to\pi^{*}\pi_{!}V=V\sqcup\Lambda V and the counit map is the map π!π(U,τ)=(UΛU,σU)(U,τ)subscript𝜋superscript𝜋𝑈𝜏square-union𝑈Λ𝑈subscript𝜎𝑈𝑈𝜏\pi_{!}\pi^{*}(U,\tau)=(U\sqcup\Lambda U,\sigma_{U})\to(U,\tau) restricting to idUsubscriptid𝑈\mathrm{id}_{U} on U𝑈U and to τ1superscript𝜏1\tau^{-1} on ΛUΛ𝑈\Lambda U. The existence of adjoints implies formally that πsuperscript𝜋\pi^{*} is continuous and cocontinuous, and that πsuperscript𝜋\pi^{*} and π!subscript𝜋\pi_{!} preserve epimorphisms. We now turn to the proof itself.

  1. (i)

    We verify Giruad’s axioms for [𝐗/C2]delimited-[]𝐗subscript𝐶2[{\mathbf{X}}/C_{2}]. Briefly, if {Gi}iIsubscriptsubscript𝐺𝑖𝑖𝐼\{G_{i}\}_{i\in I} is a set of generators for 𝐗𝐗{\mathbf{X}}, then {π!Gi}iIsubscriptsubscript𝜋subscript𝐺𝑖𝑖𝐼\{\pi_{!}G_{i}\}_{i\in I} is a set of generators for [𝐗/C2]delimited-[]𝐗subscript𝐶2[{\mathbf{X}}/C_{2}]. Indeed, let f,g:UV:𝑓𝑔𝑈𝑉f,g:U\to V be distinct morphisms in [𝐗/C2]delimited-[]𝐗subscript𝐶2[{\mathbf{X}}/C_{2}]. Since πsuperscript𝜋\pi^{*} is faithful, πf,πg:πUπV:superscript𝜋𝑓superscript𝜋𝑔superscript𝜋𝑈superscript𝜋𝑉\pi^{*}f,\pi^{*}g:\pi^{*}U\to\pi^{*}V are distinct in 𝐗𝐗{\mathbf{X}}, hence there exist iI𝑖𝐼i\in I and h:GiπUπV:subscript𝐺𝑖superscript𝜋𝑈superscript𝜋𝑉h:G_{i}\to\pi^{*}U\to\pi^{*}V such that πfhπghsuperscript𝜋𝑓superscript𝜋𝑔\pi^{*}f\circ h\neq\pi^{*}g\circ h. By the adjunction between π!subscript𝜋\pi_{!} and πsuperscript𝜋\pi^{*}, the morphism hh corresponds to a morphism h:π!GiU:superscriptsubscript𝜋subscript𝐺𝑖𝑈h^{\prime}:\pi_{!}G_{i}\to U in [𝐗/C2]delimited-[]𝐗subscript𝐶2[{\mathbf{X}}/C_{2}] such that fhgh𝑓superscript𝑔superscriptf\circ h^{\prime}\neq g\circ h^{\prime}, as required.

    That sums are disjoint and equivalence relations are effective in [𝐗/C2]delimited-[]𝐗subscript𝐶2[{\mathbf{X}}/C_{2}] can be checked with the help of the forgetful functor πsuperscript𝜋\pi^{*} and the fact that these properties hold in 𝐗𝐗{\mathbf{X}}. Finally, the existence of colimits and the fact that they commute with fiber products can be checked directly.

  2. (ii)

    This is immediate from the adjunctions between π!subscript𝜋\pi_{!}, πsuperscript𝜋\pi^{*} and πsubscript𝜋\pi_{*} noted above.

  3. (iii)

    We may replace {(Ui,τi)}iIsubscriptsubscript𝑈𝑖subscript𝜏𝑖𝑖𝐼\{(U_{i},\tau_{i})\}_{i\in I} with their disjoint union, denoted (U,τ)superscript𝑈superscript𝜏(U^{\prime},\tau^{\prime}), to assume that I𝐼I consists of a single element. We need to show that UUsuperscript𝑈𝑈U^{\prime}\to U is an epimorphism in 𝐗𝐗{\mathbf{X}} if and only if (U,τ)(U,τ)superscript𝑈superscript𝜏𝑈𝜏(U^{\prime},\tau^{\prime})\to(U,\tau) is an epimorphism in [𝐗/C2]delimited-[]𝐗subscript𝐶2[{\mathbf{X}}/C_{2}]. The “if” part follows from the fact that πsuperscript𝜋\pi^{*} preserves epimorphisms, being a left adjoint. To see the converse, it is enough to show that the composition π!U=π!π(U,τ)counit(U,τ)(U,τ)subscript𝜋superscript𝑈subscript𝜋superscript𝜋superscript𝑈superscript𝜏counitsuperscript𝑈superscript𝜏𝑈𝜏\pi_{!}U^{\prime}=\pi_{!}\pi^{*}(U^{\prime},\tau^{\prime})\xrightarrow{\text{counit}}(U^{\prime},\tau^{\prime})\to(U,\tau) is an epimorphism. Let (V,σ)[𝐗/C2]𝑉𝜎delimited-[]𝐗subscript𝐶2(V,\sigma)\in[{\mathbf{X}}/C_{2}]. Then Hom[𝐗/C2](π!U,(V,σ))Hom𝐗(U,π(V,σ))=Hom𝐗(U,V)subscriptHomdelimited-[]𝐗subscript𝐶2subscript𝜋superscript𝑈𝑉𝜎subscriptHom𝐗superscript𝑈superscript𝜋𝑉𝜎subscriptHom𝐗superscript𝑈𝑉\operatorname{Hom}_{[{\mathbf{X}}/C_{2}]}(\pi_{!}U^{\prime},(V,\sigma))\cong\operatorname{Hom}_{{\mathbf{X}}}(U^{\prime},\pi^{*}(V,\sigma))=\operatorname{Hom}_{{\mathbf{X}}}(U^{\prime},V), and under this isomorphism the map

    Hom[𝐗/C2]((U,τ),(V,σ))Hom[𝐗/C2](π!U,(V,σ))subscriptHomdelimited-[]𝐗subscript𝐶2𝑈𝜏𝑉𝜎subscriptHomdelimited-[]𝐗subscript𝐶2subscript𝜋superscript𝑈𝑉𝜎\operatorname{Hom}_{[{\mathbf{X}}/C_{2}]}((U,\tau),(V,\sigma))\to\operatorname{Hom}_{[{\mathbf{X}}/C_{2}]}(\pi_{!}U^{\prime},(V,\sigma))

    is the composition Hom[𝐗/C2]((U,τ),(V,σ))Hom𝐗(U,V)Hom𝐗(U,V)subscriptHomdelimited-[]𝐗subscript𝐶2𝑈𝜏𝑉𝜎subscriptHom𝐗𝑈𝑉subscriptHom𝐗superscript𝑈𝑉\operatorname{Hom}_{[{\mathbf{X}}/C_{2}]}((U,\tau),(V,\sigma))\hookrightarrow\operatorname{Hom}_{{\mathbf{X}}}(U,V)\to\operatorname{Hom}_{{\mathbf{X}}}(U^{\prime},V), which is injective since UUsuperscript𝑈𝑈U^{\prime}\to U an epimorphism. Thus, (U,τ)(U,τ)superscript𝑈superscript𝜏𝑈𝜏(U^{\prime},\tau^{\prime})\to(U,\tau) is an epimorphism.

  4. (iv)

    Let {ri}iIsubscriptsubscript𝑟𝑖𝑖𝐼\{r_{i}\}_{i\in I} be (U,τ)𝑈𝜏(U,\tau)-sections of (𝒪𝐗,λ)subscript𝒪𝐗𝜆({\mathcal{O}}_{\mathbf{X}},\lambda) generating the unit ideal. Then {πri}iIsubscriptsuperscript𝜋subscript𝑟𝑖𝑖𝐼\{\pi^{*}r_{i}\}_{i\in I} generate the unit ideal in 𝒪𝐗(U)subscript𝒪𝐗𝑈{\mathcal{O}}_{\mathbf{X}}(U), and hence there exists a covering {αi:UiU}iIsubscriptconditional-setsubscript𝛼𝑖subscript𝑈𝑖𝑈𝑖𝐼\{\alpha_{i}:U_{i}\to U\}_{i\in I} such that ri𝒪𝐗(Ui)×subscript𝑟𝑖subscript𝒪𝐗superscriptsubscript𝑈𝑖r_{i}\in{{\mathcal{O}}_{\mathbf{X}}(U_{i})^{\times}} for all i𝑖i. (We remark that 𝒪𝐗()subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}}(\emptyset) is the 00-ring, in which the unique element is invertible; it is possible that some of the Uisubscript𝑈𝑖U_{i} called for in this covering are initial objects.)

    Fix iI𝑖𝐼i\in I. The adjunction between π!subscript𝜋\pi_{!} and πsuperscript𝜋\pi^{*} gives rise to a morphism βi:π!Ui(U,τ):subscript𝛽𝑖subscript𝜋subscript𝑈𝑖𝑈𝜏\beta_{i}:\pi_{!}U_{i}\to(U,\tau), adjoint to αi:UiU=π(U,τ):subscript𝛼𝑖subscript𝑈𝑖𝑈superscript𝜋𝑈𝜏\alpha_{i}:U_{i}\to U=\pi^{*}(U,\tau), and an isomorphism of rings 𝒪𝐗(Ui)=Hom𝐗(Ui,𝒪𝐗)Hom[𝐗/C2](π!Ui,(𝒪𝐗,λ))=𝒪[𝐗/C2](π!Ui)subscript𝒪𝐗subscript𝑈𝑖subscriptHom𝐗subscript𝑈𝑖subscript𝒪𝐗subscriptHomdelimited-[]𝐗subscript𝐶2subscript𝜋subscript𝑈𝑖subscript𝒪𝐗𝜆subscript𝒪delimited-[]𝐗subscript𝐶2subscript𝜋subscript𝑈𝑖{\mathcal{O}}_{\mathbf{X}}(U_{i})=\operatorname{Hom}_{\mathbf{X}}(U_{i},{\mathcal{O}}_{\mathbf{X}})\cong\operatorname{Hom}_{[{\mathbf{X}}/C_{2}]}(\pi_{!}U_{i},({\mathcal{O}}_{\mathbf{X}},\lambda))={\mathcal{O}}_{[{\mathbf{X}}/C_{2}]}(\pi_{!}U_{i}). Unfolding the definitions, one finds that under this isomorphism, ri|π!Ui=riβi𝒪[𝐗/C2](π!Ui)evaluated-atsubscript𝑟𝑖subscript𝜋subscript𝑈𝑖subscript𝑟𝑖subscript𝛽𝑖subscript𝒪delimited-[]𝐗subscript𝐶2subscript𝜋subscript𝑈𝑖r_{i}|_{\pi_{!}U_{i}}=r_{i}\circ\beta_{i}\in{\mathcal{O}}_{[{\mathbf{X}}/C_{2}]}(\pi_{!}U_{i}) corresponds to πri|Ui=πriαievaluated-atsuperscript𝜋subscript𝑟𝑖subscript𝑈𝑖superscript𝜋subscript𝑟𝑖subscript𝛼𝑖\pi^{*}r_{i}|_{U_{i}}=\pi^{*}r_{i}\circ\alpha_{i}, which is invertible. Thus, risubscript𝑟𝑖r_{i} is invertible in 𝒪[𝐗/C2](π!Ui)subscript𝒪delimited-[]𝐗subscript𝐶2subscript𝜋subscript𝑈𝑖{\mathcal{O}}_{[{\mathbf{X}}/C_{2}]}(\pi_{!}U_{i}). By (iii), the collection {βi:π!Ui(U,τ)}iIsubscriptconditional-setsubscript𝛽𝑖subscript𝜋subscript𝑈𝑖𝑈𝜏𝑖𝐼\{\beta_{i}:\pi_{!}U_{i}\to(U,\tau)\}_{i\in I} is a covering, so we have shown that [𝐗/C2]delimited-[]𝐗subscript𝐶2[{\mathbf{X}}/C_{2}] is locally ringed by 𝒪[𝐗/C2]subscript𝒪delimited-[]𝐗subscript𝐶2{\mathcal{O}}_{[{\mathbf{X}}/C_{2}]}.

  5. (v)

    One checks that πλ:π𝒪𝐗π𝒪𝐗:subscript𝜋𝜆subscript𝜋subscript𝒪𝐗subscript𝜋subscript𝒪𝐗\pi_{*}\lambda:\pi_{*}{\mathcal{O}}_{\mathbf{X}}\to\pi_{*}{\mathcal{O}}_{\mathbf{X}} is the morphism

    (𝒪𝐗×Λ𝒪𝐗,τ𝒪𝐗)(x,y)(yλ,xλ)(𝒪𝐗×Λ𝒪𝐗,τ𝒪𝐗),maps-to𝑥𝑦superscript𝑦𝜆superscript𝑥𝜆subscript𝒪𝐗Λsubscript𝒪𝐗subscript𝜏subscript𝒪𝐗subscript𝒪𝐗Λsubscript𝒪𝐗subscript𝜏subscript𝒪𝐗({\mathcal{O}}_{\mathbf{X}}\times\Lambda{\mathcal{O}}_{\mathbf{X}},\tau_{{\mathcal{O}}_{\mathbf{X}}})\xrightarrow{(x,y)\mapsto(y^{\lambda},x^{\lambda})}({\mathcal{O}}_{\mathbf{X}}\times\Lambda{\mathcal{O}}_{\mathbf{X}},\tau_{{\mathcal{O}}_{\mathbf{X}}}),

    and so 𝒪[𝐗/C2]subscript𝒪delimited-[]𝐗subscript𝐶2{\mathcal{O}}_{[{\mathbf{X}}/C_{2}]} is the equalizer of id,πλ:π𝒪𝐗π𝒪𝐗:idsubscript𝜋𝜆subscript𝜋subscript𝒪𝐗subscript𝜋subscript𝒪𝐗\mathrm{id},\pi_{*}\lambda:\pi_{*}{\mathcal{O}}_{\mathbf{X}}\to\pi_{*}{\mathcal{O}}_{\mathbf{X}}. That πsubscript𝜋\pi_{*} preserves epimorphisms can be checked directly using the definitions and (iii). The verification of the coherence conditions in Definition 4.2.1 is routine. ∎

4.5. Ramification

Let 𝐗𝐗{\mathbf{X}} be a locally ringed topos with an involution λ=(Λ,ν,λ)𝜆Λ𝜈𝜆\lambda=(\Lambda,\nu,\lambda), and let π:𝐗𝐘:𝜋𝐗𝐘\pi:{\mathbf{X}}\to{\mathbf{Y}} be an exact quotient, see Definition 4.3.1. For brevity, write

R=π𝒪𝐗andS=𝒪𝐘.formulae-sequence𝑅subscript𝜋subscript𝒪𝐗and𝑆subscript𝒪𝐘R=\pi_{*}{\mathcal{O}}_{\mathbf{X}}\qquad\text{and}\qquad S={\mathcal{O}}_{\mathbf{Y}}\ .

As in Subsection 4.3, we write πλ:RπΛ𝒪𝐗=R:subscript𝜋𝜆𝑅subscript𝜋Λsubscript𝒪𝐗𝑅\pi_{*}\lambda:R\to\pi_{*}\Lambda{\mathcal{O}}_{\mathbf{X}}=R as λ𝜆\lambda. The automorphism λ:RR:𝜆𝑅𝑅\lambda:R\to R is an involution the fixed ring of which is S𝑆S.

Definition 4.5.1.

Let V𝑉V be an object of 𝐘𝐘{\mathbf{Y}}. We say that π𝜋\pi is unramified (relative to λ𝜆\lambda) along V𝑉V if RVsubscript𝑅𝑉R_{V} is a quadratic étale SVsubscript𝑆𝑉S_{V}-algebra in 𝐘/V𝐘𝑉{\mathbf{Y}}/V, see Subsection 3.2. Otherwise, π𝜋\pi is said to be ramified along V𝑉V.

The morphism π𝜋\pi is said to be unramified if it is unramified along 𝐘subscript𝐘*_{\mathbf{Y}}, and ramified otherwise. It is everywhere ramified if π𝜋\pi is ramified along every non-initial object of 𝐘𝐘{\mathbf{Y}}.

We alert the reader that, contrary to the common use of the term “ramification”, we consider trivial C2subscript𝐶2C_{2}-quotients as everywhere ramified.

Example 4.5.2.

Suppose (𝐗,𝒪𝐗)𝐗subscript𝒪𝐗({\mathbf{X}},{\mathcal{O}}_{\mathbf{X}}) is given a weakly trivial involution and π𝜋\pi is the trivial C2subscript𝐶2C_{2}-quotient, namely, the identity map id:(𝐗,𝒪𝐗)(𝐗,𝒪𝐗):id𝐗subscript𝒪𝐗𝐗subscript𝒪𝐗\mathrm{id}:({\mathbf{X}},{\mathcal{O}}_{\mathbf{X}})\to({\mathbf{X}},{\mathcal{O}}_{\mathbf{X}}) (Example 4.3.5). Then π𝜋\pi is everywhere ramified. Indeed, in this case R=S=𝒪𝐗𝑅𝑆subscript𝒪𝐗R=S={\mathcal{O}}_{\mathbf{X}} and and λ=idR𝜆subscriptid𝑅\lambda=\mathrm{id}_{R}. Since 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}} is a local ring object, for any V𝑉V\ncong\emptyset in 𝐗𝐗{\mathbf{X}}, the ring 𝒪𝐗(V)subscript𝒪𝐗𝑉{\mathcal{O}}_{\mathbf{X}}(V) is nonzero, and so RVsubscript𝑅𝑉R_{V} cannot be locally free of rank 222 over SVsubscript𝑆𝑉S_{V}.

For any object V𝑉V of 𝐘𝐘{\mathbf{Y}}, define U(V)𝑈𝑉U(V) to be a singleton if π𝜋\pi is unramified along V𝑉V, and an empty set otherwise. It follows from Lemma 3.2.2 that VU(V)maps-to𝑉𝑈𝑉V\mapsto U(V) defines a sheaf (the action of U𝑈U on morphisms in 𝐘𝐘{\mathbf{Y}} is uniquely determined), which is then represented by an object of 𝐘𝐘{\mathbf{Y}}, denoted U=Uπ𝑈subscript𝑈𝜋U=U_{\pi}. We call U𝑈U the unbranched locus of π𝜋\pi. It is a subobject of 𝐘subscript𝐘*_{\mathbf{Y}}. Clearly, π𝜋\pi is unramified if and only if U=𝐘𝑈subscript𝐘U=*_{\mathbf{Y}}, and π𝜋\pi is everywhere ramified if and only if U=𝐘𝑈subscript𝐘U=\emptyset_{\mathbf{Y}}.

The following propositions give a more concrete description of the unbranched locus when π:𝐗𝐘:𝜋𝐗𝐘\pi:{\mathbf{X}}\to{\mathbf{Y}} is induced by a C2subscript𝐶2C_{2}-quotient of schemes or topological spaces, see Examples 4.3.3 and 4.3.4.

Proposition 4.5.3.

In the situation of Example 4.3.3, i.e., when π:𝐗𝐘:𝜋𝐗𝐘\pi:{\mathbf{X}}\to{\mathbf{Y}} is obtained from a good C2subscript𝐶2C_{2}-quotient of schemes π:XY:𝜋𝑋𝑌\pi:X\to Y by taking étale ringed topoi, the unbranched locus of π𝜋\pi, defined above, is represented by an open subscheme UY𝑈𝑌U\subseteq Y. The latter can be defined in any of the following equivalent ways:

  1. (a)

    U𝑈U is the largest open subset of Y𝑌Y such that πU:π1(U)U:subscript𝜋𝑈superscript𝜋1𝑈𝑈\pi_{U}:\pi^{-1}(U)\to U is quadratic étale.

  2. (b)

    The (Zariski) points of U𝑈U are those points yY𝑦𝑌y\in Y such that X×YSpec𝒪Y,ySpec𝒪Y,ysubscript𝑌𝑋Specsubscript𝒪𝑌𝑦Specsubscript𝒪𝑌𝑦X\times_{Y}\operatorname{Spec}{\mathcal{O}}_{Y,y}\to\operatorname{Spec}{\mathcal{O}}_{Y,y} is quadratic étale.

  3. (c)

    The (Zariski) points of U𝑈U are those points yY𝑦𝑌y\in Y such that X×YSpec𝒪Y,yshSpec𝒪Y,yshsubscript𝑌𝑋Specsuperscriptsubscript𝒪𝑌𝑦shSpecsuperscriptsubscript𝒪𝑌𝑦shX\times_{Y}\operatorname{Spec}{\mathcal{O}}_{Y,y}^{\mathrm{sh}}\to\operatorname{Spec}{\mathcal{O}}_{Y,y}^{\mathrm{sh}} is quadratic étale; here, 𝒪Y,yshsuperscriptsubscript𝒪𝑌𝑦sh{\mathcal{O}}_{Y,y}^{\mathrm{sh}} is the strict henselization of 𝒪Y,ysubscript𝒪𝑌𝑦{\mathcal{O}}_{Y,y}.

  4. (d)

    The (Zariski) points of YU𝑌𝑈Y-U are those points yY𝑦𝑌y\in Y such that the set π1(y)superscript𝜋1𝑦\pi^{-1}(y) is a singleton {x}𝑥\{x\}, and λ𝜆\lambda induces the identity on k(x)𝑘𝑥k(x).

Consequently, π:𝐗𝐘:𝜋𝐗𝐘\pi:{\mathbf{X}}\to{\mathbf{Y}} is unramified if and only if π:XY:𝜋𝑋𝑌\pi:X\to Y is quadratic étale.

Proof.

By virtue of Lemma 3.2.2, if VY𝑉𝑌V\to Y is an étale morphism having image U𝑈U in Y𝑌Y, then πV:X×YVV:subscript𝜋𝑉subscript𝑌𝑋𝑉𝑉\pi_{V}:X\times_{Y}V\to V is quadratic étale if and only if πU:π1(U)U:subscript𝜋𝑈superscript𝜋1𝑈𝑈\pi_{U}:\pi^{-1}(U)\to U is quadratic étale. It follows that there exists a maximal open subset U𝑈U of Y𝑌Y with the property that πUsubscript𝜋𝑈\pi_{U} is quadratic étale, and any VX𝑉𝑋V\to X as above factors through the inclusion UX𝑈𝑋U\subseteq X. We also let U𝑈U denote the sheaf it represents in 𝐘=Sh(Yét)𝐘Shsubscript𝑌ét{\mathbf{Y}}=\text{\bf Sh}(Y_{\text{\'{e}t}}).

Since U𝑈U is a subobject of 𝐘subscript𝐘*_{\mathbf{Y}}, the set U(V)𝑈𝑉U(V) is a singleton or empty for all V𝐘𝑉𝐘V\in{\mathbf{Y}}. To show that U𝑈U is the unbranched locus of π:𝐗𝐘:𝜋𝐗𝐘\pi:{\mathbf{X}}\to{\mathbf{Y}}, it is enough to show that π𝜋\pi is unramified along an object V𝑉V of 𝐘𝐘{\mathbf{Y}} if and only if there exists a morphism VU𝑉𝑈V\to U. For any such V𝑉V, we can find a covering {ViV}isubscriptsubscript𝑉𝑖𝑉𝑖\{V_{i}\to V\}_{i} with each Visubscript𝑉𝑖V_{i} represented by some (V~iY)subscript~𝑉𝑖𝑌(\tilde{V}_{i}\to Y) in Yétsubscript𝑌étY_{\text{\'{e}t}}. By Lemma 3.2.2, π𝜋\pi unramified along V𝑉V if and only if π𝜋\pi is unramified along each Visubscript𝑉𝑖V_{i}. Furthermore, if for each iI𝑖𝐼i\in I there is a morphism ViUsubscript𝑉𝑖𝑈V_{i}\to U (in 𝐘𝐘{\mathbf{Y}}), then these morphisms must patch to a morphism VU𝑉𝑈V\to U, because U(Vi×VVj)𝑈subscript𝑉subscript𝑉𝑖subscript𝑉𝑗U(V_{i}\times_{V}V_{j}) is a singleton. It is therefore enough to show that π𝜋\pi is unramified along Visubscript𝑉𝑖V_{i} if and only if there exists a morphism V~iUsubscript~𝑉𝑖𝑈\tilde{V}_{i}\to U in Yétsubscript𝑌étY_{\text{\'{e}t}}. The latter holds precisely when im(V~iY)Uimsubscript~𝑉𝑖𝑌𝑈\operatorname{im}(\tilde{V}_{i}\to Y)\subseteq U, and so the claim follows from the definition of U𝑈U.

We finish by showing that the different characterizations of U𝑈U are equivalent. The equivalence of (a) and (b) follows from Corollary 3.3.5, and the equivalence of (b) and (d) follows from Theorem 3.3.8. That the condition in (b) implies the condition in (c) is clear. It remains to prove the converse. Since 𝒪Y,yshsuperscriptsubscript𝒪𝑌𝑦sh{\mathcal{O}}_{Y,y}^{\mathrm{sh}} is faithfully flat over 𝒪Y,ysubscript𝒪𝑌𝑦{\mathcal{O}}_{Y,y}, this is a consequence of Lemma 3.2.2 applied to the fpqc site of Spec𝒪Y,ySpecsubscript𝒪𝑌𝑦\operatorname{Spec}{\mathcal{O}}_{Y,y}. ∎

Proposition 4.5.4.

In the situation of Example 4.3.4, i.e., when π:𝐗𝐘:𝜋𝐗𝐘\pi:{\mathbf{X}}\to{\mathbf{Y}} is induced by a C2subscript𝐶2C_{2}-quotient of Hausdorff topological spaces π:XY=X/C2:𝜋𝑋𝑌𝑋subscript𝐶2\pi:X\to Y=X/C_{2}, the unbranched locus is represented by an open subset UY𝑈𝑌{U}\subseteq Y. Specifically, U={xX:xλx}/C2𝑈conditional-set𝑥𝑋superscript𝑥𝜆𝑥subscript𝐶2{U}=\{x\in X\,:\,x^{\lambda}\neq x\}/C_{2}. Consequently, π:𝐗𝐘:𝜋𝐗𝐘\pi:{\mathbf{X}}\to{\mathbf{Y}} is unramified if and only if C2subscript𝐶2C_{2} acts freely on X𝑋X.

Proof.

This is similar to the previous proof and is left to the reader. ∎

We refer to the situations of Examples 4.3.3 and 4.3.4 as the scheme-theoretic case and topological case, respectively. In both cases, we define the branch locus of π𝜋\pi to be the complement W:=YUassign𝑊𝑌𝑈W:=Y-U, where U𝑈U is as in Proposition 4.5.3 or Proposition 4.5.4. The ramification locus of π𝜋\pi is Z:=π1(W)assign𝑍superscript𝜋1𝑊Z:=\pi^{-1}(W). In the scheme-theoretic case, we also endow W𝑊W and Z𝑍Z with the reduced induced closed subscheme structure.

Notice that in the topological case, πU:π1(U)U:subscript𝜋𝑈superscript𝜋1𝑈𝑈\pi_{U}:\pi^{-1}(U)\to U is a double covering, while πW:π1(W)W:subscript𝜋𝑊superscript𝜋1𝑊𝑊\pi_{W}:\pi^{-1}(W)\to W is a homeomorphism. With slight modification, a similar statement holds for schemes.

Proposition 4.5.5.

In the notation of Proposition 4.5.3, let W=UY𝑊𝑈𝑌W=U-Y, and regard W𝑊W and π1(W)superscript𝜋1𝑊\pi^{-1}(W) as reduced closed subschemes of Y𝑌Y and X𝑋X, respectively. Then:

  1. (i)

    πU:π1(U)U:subscript𝜋𝑈superscript𝜋1𝑈𝑈\pi_{U}:\pi^{-1}(U)\to U is quadratic étale.

  2. (ii)

    λ:XX:𝜆𝑋𝑋\lambda:X\to X restricts to the identity morphism on π1(W)superscript𝜋1𝑊\pi^{-1}(W).

  3. (iii)

    π|π1(W):π1(W)W:evaluated-at𝜋superscript𝜋1𝑊superscript𝜋1𝑊𝑊\pi|_{\pi^{-1}(W)}:\pi^{-1}(W)\to W is a homeomorphism, and when 222 is invertible on Y𝑌Y, it is an isomorphism of schemes.

Proof.
  1. (i)

    This immediate from condition (a) in Proposition 4.5.3.

  2. (ii)

    Condition (d) of Proposition 4.5.3 implies that λ𝜆\lambda fixes the (Zariski) points of π1(W)superscript𝜋1𝑊\pi^{-1}(W). Let f𝑓f be a (Zariski) section of 𝒪π1(W)subscript𝒪superscript𝜋1𝑊{\mathcal{O}}_{\pi^{-1}(W)} and let Sπ1(W)𝑆superscript𝜋1𝑊S\subseteq\pi^{-1}(W) be the largest open subset on which ffλ𝑓superscript𝑓𝜆f-f^{\lambda} is invertible. Let sS𝑠𝑆s\in S. Then ffλ𝑓superscript𝑓𝜆f-f^{\lambda} is invertible in k(s)𝑘𝑠k(s), which is impossible by condition (d) of Proposition 4.5.3. Thus, S=𝑆S=\emptyset. Since π1(W)superscript𝜋1𝑊\pi^{-1}(W) is reduced, we conclude that ffλ=0𝑓superscript𝑓𝜆0f-f^{\lambda}=0.

  3. (iii)

    It is enough to prove the claim after restricting to an open affine covering of Y𝑌Y, so assume X=SpecA𝑋Spec𝐴X=\operatorname{Spec}A, Y=SpecB𝑌Spec𝐵Y=\operatorname{Spec}B, W=SpecB/I𝑊Spec𝐵𝐼W=\operatorname{Spec}B/I with I𝐼I a radical ideal of B𝐵B, and π1(W)=SpecA/IAsuperscript𝜋1𝑊Spec𝐴𝐼𝐴\pi^{-1}(W)=\operatorname{Spec}A/\sqrt{IA}, where IA𝐼𝐴\sqrt{IA} denotes the radical of IA𝐼𝐴IA. The morphism π|π1(W):π1(W)W:evaluated-at𝜋superscript𝜋1𝑊superscript𝜋1𝑊𝑊\pi|_{\pi^{-1}(W)}:\pi^{-1}(W)\to W is adjoint to the evident homomorphism B/IA/IA𝐵𝐼𝐴𝐼𝐴B/I\to A/\sqrt{IA}, and λ:XX:𝜆𝑋𝑋\lambda:X\to X induces an involution λ:AA:𝜆𝐴𝐴\lambda:A\to A having fixed ring B𝐵B.

    We know that π:π1(W)W:𝜋superscript𝜋1𝑊𝑊\pi:{\pi^{-1}(W)}\to W is continuous and it is a set bijection since its set-theoretic fibers consist of singletons by condition (d) of Proposition 4.5.3. Thus, proving that π|π1(W)evaluated-at𝜋superscript𝜋1𝑊\pi|_{\pi^{-1}(W)} is a homeomorphism amounts to checking that it is closed. Since any aA𝑎𝐴a\in A satisfies a2(aλ+a)a+(aλa)=0superscript𝑎2superscript𝑎𝜆𝑎𝑎superscript𝑎𝜆𝑎0a^{2}-(a^{\lambda}+a)a+(a^{\lambda}a)=0, the morphism SpecA/IASpecB/ISpec𝐴𝐼𝐴Spec𝐵𝐼\operatorname{Spec}A/\sqrt{IA}\to\operatorname{Spec}B/I is integral, and therefore closed by [de_jong_stacks_2017, Tag 01WM].

    Suppose now that 2B×2superscript𝐵2\in{B^{\times}}. We need to show that B/IA/IA𝐵𝐼𝐴𝐼𝐴B/I\to A/\sqrt{IA} is bijective. Let aA𝑎𝐴a\in A. By virtue of (ii), λ𝜆\lambda induces the identity involution on A/IA𝐴𝐼𝐴A/\sqrt{IA}, and thus a12(a+aλ)modIA𝑎modulo12𝑎superscript𝑎𝜆𝐼𝐴a\equiv\frac{1}{2}(a+a^{\lambda})\mod\sqrt{IA}. Since a+aλB𝑎superscript𝑎𝜆𝐵a+a^{\lambda}\in B, we have established the surjectivity of B/IA/IA𝐵𝐼𝐴𝐼𝐴B/I\to A/\sqrt{IA}. Next, write J=ker(B/IA/AI)𝐽kernel𝐵𝐼𝐴𝐴𝐼J=\ker(B/I\to A/\sqrt{AI}). Since SpecA/AISpecB/ISpec𝐴𝐴𝐼Spec𝐵𝐼\operatorname{Spec}A/\sqrt{AI}\to\operatorname{Spec}B/I is bijective, J𝐽J is contained in every prime ideal of B/I𝐵𝐼B/I, and since B/I𝐵𝐼B/I is reduced, J=0𝐽0J=0. ∎

Remark 4.5.6.
  1. (i)

    In Proposition 4.5.5(iii), it is in general necessary to assume that 222 is invertible in order to conclude that π:π1(W)W:𝜋superscript𝜋1𝑊𝑊\pi:\pi^{-1}(W)\to W is an isomorphism. Consider, for example, a DVR S𝑆S with 20202\neq 0 having a non-perfect residue field K𝐾K of characteristic 222, let αS×𝛼superscript𝑆\alpha\in{S^{\times}} be an element such that its image in K𝐾K is not a square, let R=S[x|x2=α]𝑅𝑆delimited-[]conditional𝑥superscript𝑥2𝛼R=S[x\,|\,x^{2}=\alpha], and let λ:RR:𝜆𝑅𝑅\lambda:R\to R be the S𝑆S-involution sending x𝑥x to x𝑥-x. Taking X=SpecR𝑋Spec𝑅X=\operatorname{Spec}R and Y=SpecS𝑌Spec𝑆Y=\operatorname{Spec}S, the set W𝑊W consists of the closed point y𝑦y of Y𝑌Y, but the induced map k(y)k(π1(y))𝑘𝑦𝑘superscript𝜋1𝑦k(y)\to k(\pi^{-1}(y)) not an isomorphism.

  2. (ii)

    Let π:XY:𝜋𝑋𝑌\pi:X\to Y be a good C2subscript𝐶2C_{2}-quotient of schemes which is everywhere ramified, and suppose 222 is invertible on Y𝑌Y. Then Proposition 4.5.5 implies that the induced morphism π:XredYred:𝜋subscript𝑋redsubscript𝑌red\pi:X_{\mathrm{red}}\to Y_{\mathrm{red}} is an isomorphism. However, in general, and in contrast to Proposition 4.5.4, it can happen that π:XY:𝜋𝑋𝑌\pi:X\to Y is not an isomorphism. For example, take X=Spec[ε]/(ε2)𝑋Specdelimited-[]𝜀superscript𝜀2X=\operatorname{Spec}\mathbb{C}[\varepsilon]/(\varepsilon^{2}) and let λ𝜆\lambda be the \mathbb{C}-involution taking ε𝜀\varepsilon to ε𝜀-\varepsilon.

Remark 4.5.7.

For a general exact C2subscript𝐶2C_{2}-quotient π:𝐗𝐘:𝜋𝐗𝐘\pi:{\mathbf{X}}\to{\mathbf{Y}} with unbranched locus U𝑈U, it is possible to define the “branch topos” 𝐖𝐖{\mathbf{W}} of π𝜋\pi as the full subcategory of 𝐘𝐘{\mathbf{Y}} consisting of objects W𝑊W such that the projection U×WU𝑈𝑊𝑈U\times W\to U is an isomorphism. In the situation of Examples 4.3.3 and 4.3.4, this turns out to give the topos of sheaves over the set-theoretic or scheme-theoretic branch locus of π:XY:𝜋𝑋𝑌\pi:X\to Y defined above. We omit the details as they will not be needed in this work.

We finish with showing that the exact quotient of Construction 4.4.10 is unramified. Thus, any locally ringed topos with involution admits an unramified exact quotient. When 𝐗𝐗{\mathbf{X}} is the étale ringed topos of a scheme X𝑋X, this generalizes the well-known fact that the morphism from X𝑋X to its quotient stack [X/C2]delimited-[]𝑋subscript𝐶2[X/C_{2}] is quadratic étale.

Proposition 4.5.8.

The exact quotient π:𝐗[𝐗/C2]:𝜋𝐗delimited-[]𝐗subscript𝐶2\pi:{\mathbf{X}}\to[{\mathbf{X}}/C_{2}] of Construction 4.4.10 is unramified.

Proof.

Recall from Construction 4.4.10 that S=(𝒪𝐗,λ)𝑆subscript𝒪𝐗𝜆S=({\mathcal{O}}_{\mathbf{X}},\lambda) and R=(𝒪𝐗×Λ𝒪𝐗,τ𝒪𝐗)𝑅subscript𝒪𝐗Λsubscript𝒪𝐗subscript𝜏subscript𝒪𝐗R=({\mathcal{O}}_{\mathbf{X}}\times\Lambda{\mathcal{O}}_{\mathbf{X}},\tau_{{\mathcal{O}}_{\mathbf{X}}}), where τ𝒪𝐗subscript𝜏subscript𝒪𝐗\tau_{{\mathcal{O}}_{\mathbf{X}}} is the interchange involution, and the morphism π#:SR:subscript𝜋#𝑆𝑅\pi_{\#}:S\to R is given by x(x,xλ)maps-to𝑥𝑥superscript𝑥𝜆x\mapsto(x,x^{\lambda}) on sections (in 𝐗𝐗{\mathbf{X}}). We shall make use of π!:𝐗[𝐗/C2]:subscript𝜋𝐗delimited-[]𝐗subscript𝐶2\pi_{!}:{\mathbf{X}}\to[{\mathbf{X}}/C_{2}], the left adjoint of πsuperscript𝜋\pi^{*} constructed in the proof of Proposition 4.4.11.

Write D:=π!(𝐗)=(Λ,σ)D:=\pi_{!}(*_{\mathbf{X}})=(*\sqcup\Lambda*,\sigma_{*}) and observe that the unique map D(𝐗,id)=[𝐗/C2]𝐷subscript𝐗idsubscriptdelimited-[]𝐗subscript𝐶2D\to(*_{\mathbf{X}},\mathrm{id})=*_{[{\mathbf{X}}/C_{2}]} is a covering by Proposition 4.4.11(iii). By Lemma 3.2.2, it is enough to show that RDsubscript𝑅𝐷R_{D} is a quadratic étale SDsubscript𝑆𝐷S_{D}-algebra. In fact, we will show that RDSD×SDsubscript𝑅𝐷subscript𝑆𝐷subscript𝑆𝐷R_{D}\cong S_{D}\times S_{D}.

We first observe that the slice category [𝐗/C2]/Ddelimited-[]𝐗subscript𝐶2𝐷[{\mathbf{X}}/C_{2}]/D is equivalent to 𝐗𝐗{\mathbf{X}}; the equivalence is given by mapping (U,τ)D𝑈𝜏𝐷(U,\tau)\to D in [𝐗/C2]/Ddelimited-[]𝐗subscript𝐶2𝐷[{\mathbf{X}}/C_{2}]/D to ×(Λ)U*\times_{(*\sqcup\Lambda*)}U, and by π!subscript𝜋\pi_{!} in the other direction. Now, consider RDsubscript𝑅𝐷R_{D} and SDsubscript𝑆𝐷S_{D} as sheaves on 𝒪[𝐗/C2]/Dsubscript𝒪delimited-[]𝐗subscript𝐶2𝐷{\mathcal{O}}_{[{\mathbf{X}}/C_{2}]}/D. Then R=RDπ!superscript𝑅subscript𝑅𝐷subscript𝜋R^{\prime}=R_{D}\circ\pi_{!} and S=SDπ!superscript𝑆subscript𝑆𝐷subscript𝜋S^{\prime}=S_{D}\circ\pi_{!} are sheaves of rings on the equivalent topos 𝐗𝐗{\mathbf{X}}. Since π!subscript𝜋\pi_{!} is left adjoint to πsuperscript𝜋\pi^{*}, for all objects V𝑉V of 𝐗𝐗{\mathbf{X}}, we have natural isomorphisms of rings

R(V)superscript𝑅𝑉\displaystyle R^{\prime}(V) =R(π!V)=Hom[𝐗/C2](π!V,(𝒪𝐗×Λ𝒪𝐗,τ𝒪𝐗))𝒪𝐗(V)×Λ𝒪𝐗(V)absent𝑅subscript𝜋𝑉subscriptHomdelimited-[]𝐗subscript𝐶2subscript𝜋𝑉subscript𝒪𝐗Λsubscript𝒪𝐗subscript𝜏subscript𝒪𝐗subscript𝒪𝐗𝑉Λsubscript𝒪𝐗𝑉\displaystyle=R(\pi_{!}V)=\operatorname{Hom}_{[{\mathbf{X}}/C_{2}]}(\pi_{!}V,({\mathcal{O}}_{\mathbf{X}}\times\Lambda{\mathcal{O}}_{\mathbf{X}},\tau_{{\mathcal{O}}_{\mathbf{X}}}))\cong{\mathcal{O}}_{\mathbf{X}}(V)\times\Lambda{\mathcal{O}}_{\mathbf{X}}(V)
S(V)superscript𝑆𝑉\displaystyle S^{\prime}(V) =S(π!V)=Hom[𝐗/C2](π!V,(𝒪𝐗,λ))𝒪𝐗(V)absent𝑆subscript𝜋𝑉subscriptHomdelimited-[]𝐗subscript𝐶2subscript𝜋𝑉subscript𝒪𝐗𝜆subscript𝒪𝐗𝑉\displaystyle=S(\pi_{!}V)=\operatorname{Hom}_{[{\mathbf{X}}/C_{2}]}(\pi_{!}V,({\mathcal{O}}_{\mathbf{X}},\lambda))\cong{\mathcal{O}}_{\mathbf{X}}(V)

and under these isomorphisms, the embedding SRsuperscript𝑆superscript𝑅S^{\prime}\to R^{\prime} induced by π#:SR:subscript𝜋#𝑆𝑅\pi_{\#}:S\to R is given by x(x,xλ)maps-to𝑥𝑥superscript𝑥𝜆x\mapsto(x,x^{\lambda}) on sections. From this it follows that RS×Ssuperscript𝑅superscript𝑆superscript𝑆R^{\prime}\cong S^{\prime}\times S^{\prime} as Ssuperscript𝑆S^{\prime}-algebras, and hence RDSD×SDsubscript𝑅𝐷subscript𝑆𝐷subscript𝑆𝐷R_{D}\cong S_{D}\times S_{D} as SDsubscript𝑆𝐷S_{D}-algebras. ∎

Remark 4.5.9.

Propositions 4.5.3 and 4.5.4 provide plenty of examples where a locally ringed topos with involution admits a ramified exact quotient. However, Proposition 4.5.8 shows that these examples also admit unramified exact quotients. It follows that exact quotients are not unique in general.

5. Classifying Involutions into Types

The purpose of this section is to classify involutions of Azumaya algebras into types in such a way which generalizes the familiar classification of involutions of central simple algebras over fields as orthogonal, symplectic or unitary.

Throughout, 𝐗𝐗{\mathbf{X}} denotes a locally ringed topos with an involution λ=(Λ,ν,λ)𝜆Λ𝜈𝜆\lambda=(\Lambda,\nu,\lambda) and π:𝐗𝐘:𝜋𝐗𝐘\pi:{\mathbf{X}}\to{\mathbf{Y}} is an exact quotient relative to λ𝜆\lambda, see Definition 4.3.1. For brevity, we write

R=π𝒪𝐗andS=𝒪𝐘,formulae-sequence𝑅subscript𝜋subscript𝒪𝐗and𝑆subscript𝒪𝐘R=\pi_{*}{\mathcal{O}}_{\mathbf{X}}\qquad\text{and}\qquad S={\mathcal{O}}_{\mathbf{Y}},

and, abusing the notation, denote the involution πλ:RR:subscript𝜋𝜆𝑅𝑅\pi_{*}\lambda:R\to R by λ𝜆\lambda. Theorem 4.3.11 and Corollary 4.3.14 imply that Azumaya 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}}-algebras with a λ𝜆\lambda-involution are equivalent to Azumaya R𝑅R-algebras with a λ𝜆\lambda-involution, and the latter are easier to work with.

In fact, most of the results of this section can be phrased with no direct reference to 𝐗𝐗{\mathbf{X}} or the quotient map π𝜋\pi, assuming only a locally ringed topos (𝐘,S)𝐘𝑆({\mathbf{Y}},S), an S𝑆S-algebra R𝑅R, and an involution λ:RR:𝜆𝑅𝑅\lambda:R\to R having fixed ring S𝑆S.

We remind the reader that Azumaya 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}}-algebras are always assumed to have a degree which is fixed by ΛΛ\Lambda, see Remark 4.3.13. This is automatic when 𝐗𝐗{\mathbf{X}} is connected.

5.1. Types of Involutions

Suppose K𝐾K is a field and λ:KK:𝜆𝐾𝐾\lambda:K\to K is an involution, the fixed subfield of which is F𝐹F. Classically, when λ=idK𝜆subscriptid𝐾\lambda=\mathrm{id}_{K}, the λ𝜆\lambda-involutions of central simple K𝐾K-algebras are divided into two types, orthogonal and symplectic, whereas in the case λidK𝜆subscriptid𝐾\lambda\neq\mathrm{id}_{K}, they are simply called unitary; see [knus_book_1998-1, §2]. This classification satisfies the following two properties:

  1. (i)

    If (A,τ)𝐴𝜏(A,\tau) and (A,τ)superscript𝐴superscript𝜏(A^{\prime},\tau^{\prime}) are central simple K𝐾K-algebras with λ𝜆\lambda-involutions such that degA=degAdeg𝐴degsuperscript𝐴\operatorname{deg}A=\operatorname{deg}A^{\prime}, then τ𝜏\tau and τsuperscript𝜏\tau^{\prime} are of the same type if and only if (A,τ)𝐴𝜏(A,\tau) and (A,τ)superscript𝐴superscript𝜏(A^{\prime},\tau^{\prime}) become isomorphic as algebras with involution over an algebraic closure of F𝐹F.

  2. (ii)

    If (A,τ)𝐴𝜏(A,\tau) is a central simple K𝐾K-algebra with λ𝜆\lambda-involution, then τ𝜏\tau has the same type as τtr:Mn×n(A)Mn×n(A):𝜏trsubscriptM𝑛𝑛𝐴subscriptM𝑛𝑛𝐴\tau\text{\rm tr}:\mathrm{M}_{n\times n}(A)\to\mathrm{M}_{n\times n}(A) given by (aij)i,j(ajiτ)i,jmaps-tosubscriptsubscript𝑎𝑖𝑗𝑖𝑗subscriptsuperscriptsubscript𝑎𝑗𝑖𝜏𝑖𝑗(a_{ij})_{i,j}\mapsto(a_{ji}^{\tau})_{i,j}.

Of these two properties, it is mainly the first that motivates the classification into types. The second property should not be disregarded as it guarantees, at least when 2K×2superscript𝐾2\in{K^{\times}}, that involutions adjoint to symmetric bilinear forms (resp. alternating bilinear forms, hermitian forms) of arbitrary rank all have the same type, see [knus_book_1998-1, §4].

Our aim in this section is to partition the λ𝜆\lambda-involutions of Azumaya R𝑅R-algebras into equivalence classes, called types, so that properties analogous to (i) and (ii) hold. To this end, we simply take the minimal equivalence relation forced by the “if” part of condition (i) and condition (ii).

Definition 5.1.1.

Let (A,τ)𝐴𝜏(A,\tau), (A,τ)superscript𝐴superscript𝜏(A^{\prime},\tau^{\prime}) be two Azumaya R𝑅R-algebras with a λ𝜆\lambda-involution. Let τtr𝜏tr\tau\text{\rm tr} denote the involution τλtrtensor-product𝜏𝜆tr\tau\otimes\lambda\text{\rm tr} of Mn×n(A)AMn×n(R)subscriptM𝑛𝑛𝐴tensor-product𝐴subscriptM𝑛𝑛𝑅\mathrm{M}_{n\times n}(A)\cong A\otimes\mathrm{M}_{n\times n}(R). On sections, it is given by (aij)i,j(ajiτ)i,jmaps-tosubscriptsubscript𝑎𝑖𝑗𝑖𝑗subscriptsuperscriptsubscript𝑎𝑗𝑖𝜏𝑖𝑗(a_{ij})_{i,j}\mapsto(a_{ji}^{\tau})_{i,j}.

We say that τ𝜏\tau and τsuperscript𝜏\tau^{\prime} are of the same λ𝜆\lambda-type or type if there exist n,n𝑛superscript𝑛n,n^{\prime}\in\mathbb{N} and a covering U𝑈U\to\ast in 𝐘𝐘\mathbf{Y} such that

(Mn×n(AU),τUtr)(Mn×n(AU),τUtr)subscriptM𝑛𝑛subscript𝐴𝑈subscript𝜏𝑈trsubscriptMsuperscript𝑛superscript𝑛subscriptsuperscript𝐴𝑈subscriptsuperscript𝜏𝑈tr(\mathrm{M}_{n\times n}(A_{U}),\tau_{U}\text{\rm tr})\cong(\mathrm{M}_{n^{\prime}\times n^{\prime}}(A^{\prime}_{U}),\tau^{\prime}_{U}\text{\rm tr})

as RUsubscript𝑅𝑈R_{U}-algebras with involution.

Being of the same λ𝜆\lambda-type is an equivalence relation. The equivalence classes will be called λ𝜆\lambda-types or just types, and the type of τ𝜏\tau will be denoted

t(τ)ort(A,τ).t𝜏ort𝐴𝜏\mathrm{t}(\tau)\qquad\text{or}\qquad\mathrm{t}(A,\tau)\ .

The set of all λ𝜆\lambda-types will be denoted Typ(λ)Typ𝜆{\mathrm{Typ}({\lambda})}. Tensor product of Azumaya algebras with involution endows Typ(λ)Typ𝜆{\mathrm{Typ}({\lambda})} with a monoid structure. We write the neutral element, represented by (R,λ)𝑅𝜆(R,\lambda), as 111.

We shall shortly see that our definition gives the familiar types in the case of fields, as well as in a number of other cases. It is no longer clear whether two involutions of the same degree and type are locally isomorphic, however, and the majority of this section will be dedicated to showing that this is indeed the case under mild assumptions. Another drawback of the definition is that it not clear how to enumerate the types it yields, and nor does it provide a way to test whether two involutions are of the same type. These problems will also be addressed, especially in the situation of Theorem 4.4.4, namely, in cases induced by a good C2subscript𝐶2C_{2}-quotient of schemes on which 222 is invertible, or by a C2subscript𝐶2C_{2}-quotient of Hausdorff topological spaces.

Remark 5.1.2.

Let (A,τ)𝐴𝜏(A,\tau), (A,τ)superscript𝐴superscript𝜏(A^{\prime},\tau^{\prime}) be two Azumaya 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}}-algebras in 𝐗𝐗\mathbf{X} with a λ𝜆\lambda-involution. Using Corollary 4.3.14, we say that τ𝜏\tau and τsuperscript𝜏\tau^{\prime} have the same λ𝜆\lambda-type (relative to π𝜋\pi) when the same holds after applying πsubscript𝜋\pi_{*}. The equivalence class of (A,τ)𝐴𝜏(A,\tau) is denoted tπ(τ)subscriptt𝜋𝜏\mathrm{t}_{\pi}(\tau) or tπ(A,τ)subscriptt𝜋𝐴𝜏\mathrm{t}_{\pi}(A,\tau) and the monoid of types is denoted Typπ(λ)subscriptTyp𝜋𝜆{\mathrm{Typ}_{\pi}({\lambda})}.

We warn the reader that the λ𝜆\lambda-type of a λ𝜆\lambda-involution of an Azumaya 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}}-algebra depends on the choice of the quotient π:𝐗𝐘:𝜋𝐗𝐘\pi:\mathbf{X}\to\mathbf{Y}, which is why we include π𝜋\pi in the notation.

For instance, we shall see below in Theorem 5.2.17 that in the case where 𝐗𝐗{\mathbf{X}} is given the trivial involution and 𝐗𝐗{\mathbf{X}} is connected, then under mild assumptions, taking π𝜋\pi to be the trivial exact quotient id:𝐗𝐗:id𝐗𝐗\mathrm{id}:{\mathbf{X}}\to{\mathbf{X}} results in two λ𝜆\lambda-types, whereas taking π𝜋\pi to be the exact quotient 𝐗[𝐗/C2]𝐗delimited-[]𝐗subscript𝐶2{\mathbf{X}}\to[{\mathbf{X}}/C_{2}] of Construction 4.4.10 results in only one λ𝜆\lambda-type.

Example 5.1.3.
  1. (i)

    Let X𝑋X be a connected scheme on which 222 is invertible, let λ:XX:𝜆𝑋𝑋\lambda:X\to X be the trivial involution and let Y:=X/C2=Xassign𝑌𝑋subscript𝐶2𝑋Y:=X/C_{2}=X. Consider the exact quotient obtained from π:XY:𝜋𝑋𝑌\pi:X\to Y by taking étale ringed topoi, see Example 4.3.3. In this case, 𝐗=𝐘=Sh(Xét)𝐗𝐘Shsubscript𝑋ét\mathbf{X}=\mathbf{Y}=\text{\bf Sh}(X_{\text{\'{e}t}}), R=S=𝒪X𝑅𝑆subscript𝒪𝑋R=S=\mathcal{O}_{X} and λ=idR𝜆subscriptid𝑅\lambda=\mathrm{id}_{R}. Thus, an Azumaya R𝑅R-algebra with a λ𝜆\lambda-involution is simply an Azumaya 𝒪Xsubscript𝒪𝑋{\mathcal{O}}_{X}-algebra with an involution of the first kind on X𝑋X. It is well known that there are two λ𝜆\lambda-types: orthogonal and symplectic. The orthogonal type is represented by (R,idR)𝑅subscriptid𝑅(R,\mathrm{id}_{R}) and the symplectic type is represented by (M2×2(R),sp)subscriptM22𝑅sp(\mathrm{M}_{2\times 2}(R),{\mathrm{sp}}), where spsp{\mathrm{sp}} is given by xh2xtrh21maps-to𝑥subscript2superscript𝑥trsuperscriptsubscript21x\mapsto h_{2}x^{\text{\rm tr}}h_{2}^{-1} on sections and h2=[0110]subscript2delimited-[]0110h_{2}=[\begin{smallmatrix}0&1\\ -1&0\end{smallmatrix}]. Moreover, every Azumaya R𝑅R-algebra of degree n𝑛n with an involution of the first kind is locally isomorphic either to (Mn(R),tr)subscriptM𝑛𝑅tr(\mathrm{M}_{n}(R),\text{\rm tr}) or to (Mn(R),sp)subscriptM𝑛𝑅sp(\mathrm{M}_{n}(R),{\mathrm{sp}}), where in the latter case n=2m𝑛2𝑚n=2m and spsp{\mathrm{sp}} is given sectionwise by xhnxtrhn1maps-to𝑥subscript𝑛superscript𝑥trsuperscriptsubscript𝑛1x\mapsto h_{n}x^{\text{\rm tr}}h_{n}^{-1} with hn=[0ImIm0]subscript𝑛delimited-[]0subscript𝐼𝑚subscript𝐼𝑚0h_{n}=[\begin{smallmatrix}0&I_{m}\\ -I_{m}&0\end{smallmatrix}]; see [knus_quadratic_1991, III.§8.5] or [parimala_92, §1.1]. In this case, Typ(λ)Typ𝜆{\mathrm{Typ}({\lambda})} is isomorphic to the group {±1}plus-or-minus1\{\pm 1\}.

  2. (ii)

    Let π:XY:𝜋𝑋𝑌\pi:X\to Y be a quadratic étale morphism, and let λ:XY:𝜆𝑋𝑌\lambda:X\to Y denote the canonical Y𝑌Y-involution of X𝑋X. Again, let π:𝐗𝐘:𝜋𝐗𝐘\pi:{\mathbf{X}}\to{\mathbf{Y}} denote the exact quotient obtained from π:XY:𝜋𝑋𝑌\pi:X\to Y by taking étale ringed topoi. In this case, R=π𝒪X𝑅subscript𝜋subscript𝒪𝑋R=\pi_{*}{\mathcal{O}}_{X} is quadratic étale over S=𝒪Y𝑆subscript𝒪𝑌S={\mathcal{O}}_{Y}, and λ𝜆\lambda-involutions are known as unitary involutions. There is only one type in this situation, and moreover, any Azumaya R𝑅R-algebra of degree n𝑛n with a λ𝜆\lambda-involution is locally isomorphic to (Mn(R),λtr)subscriptM𝑛𝑅𝜆tr(\mathrm{M}_{n}(R),\lambda\text{\rm tr}); these well-known facts can be found in [parimala_92, §1.2] without proof, but they follow from the results in the sequel. We were unable to find a source providing complete proofs.

Example 5.1.4.

Let K𝐾K be a perfect field of characteristic 222, and consider the case of the trivial involution on K𝐾K. As in Example 5.1.3(i), this corresponds to taking 𝐗=𝐘=Sh((SpecK)ét)𝐗𝐘ShsubscriptSpec𝐾ét\mathbf{X}=\mathbf{Y}=\text{\bf Sh}((\operatorname{Spec}K)_{\text{\'{e}t}}), R=S=𝒪SpecK𝑅𝑆subscript𝒪Spec𝐾R=S={\mathcal{O}}_{\operatorname{Spec}K} and λ=id𝜆id\lambda=\mathrm{id}. Azumaya R𝑅R-algebras with a λ𝜆\lambda-involution are therefore central simple K𝐾K-algebras with an involution of the first kind. There are again two types in this case, again called orthogonal and symplectic, but Typ(λ)Typ𝜆{\mathrm{Typ}({\lambda})} is isomorphic to the multiplicative monoid {0,1}01\{0,1\} with 00 corresponding to the symplectic type; see [knus_book_1998-1, §2]. This shows that the theory in characteristic 222 is substantially different from that in other characteristics.

The assumption that K𝐾K is perfect can be dropped if one replaces the étale site with the fppf site (consult Remark 4.4.9).

5.2. Coarse types

In Subsection 5.1, we defined the type of a λ𝜆\lambda-involution of an Azumaya R𝑅R-algebra in terms of the entire class of algebras and not as an intrinsic invariant of the involution. We now introduce another invariant of λ𝜆\lambda-involutions, called the coarse type, which, while in general coarser than the type, will enjoy an intrinsic definition. It will turn out that under mild assumptions the invariants are equivalent, and this will be used to address the questions raised in Subsection 5.1. Apart from that, coarse types will also be needed in proving the main results of Section 6.

We begin by defining the abelian group object N𝑁N of 𝐘𝐘{\mathbf{Y}} via the exact sequence

(3) 1NR×xxλxS×1𝑁superscript𝑅maps-to𝑥superscript𝑥𝜆𝑥superscript𝑆1\to N\to{R^{\times}}\xrightarrow{x\mapsto x^{\lambda}x}{S^{\times}}

and the abelian group object T𝑇T of 𝐘𝐘{\mathbf{Y}} via the short exact sequence

(4) 1R×/S×xxλx1NT1.1superscript𝑅superscript𝑆maps-to𝑥superscript𝑥𝜆superscript𝑥1𝑁𝑇11\to{R^{\times}}/{S^{\times}}\xrightarrow{x\mapsto x^{\lambda}x^{-1}}N\to T\to 1\ .

The group N𝑁N should be regarded as the group of elements of λ𝜆\lambda-norm 111. We call the global sections of T𝑇T coarse λ𝜆\lambda-types and write

cTyp(λ)=H0(T).cTyp𝜆superscriptH0𝑇{\mathrm{cTyp}({\lambda})}=\mathrm{H}^{0}(T)\ .

The following example and propositions give some hints about the structure of T𝑇T.

Example 5.2.1.

If the involution of 𝐗𝐗{\mathbf{X}} is trivial and the quotient map is the identity, then N=μ2,R𝑁subscript𝜇2𝑅N=\mu_{2,R} and the map xx1xλ:R×/S×N:maps-to𝑥superscript𝑥1superscript𝑥𝜆superscript𝑅superscript𝑆𝑁x\mapsto x^{-1}x^{\lambda}:{R^{\times}}/{S^{\times}}\to N is trivial, hence T=μ2,R𝑇subscript𝜇2𝑅T=\mu_{2,R} and cTyp(λ)=H0(μ2,R)={xH0(R):x2=1}cTyp𝜆superscriptH0subscript𝜇2𝑅conditional-set𝑥superscriptH0𝑅superscript𝑥21{\mathrm{cTyp}({\lambda})}=\mathrm{H}^{0}(\mu_{2,R})=\{x\in\mathrm{H}^{0}(R)\,:\,x^{2}=1\}.

Proposition 5.2.2.

If π𝜋\pi is unramified along an object U𝑈U of 𝐘𝐘{\mathbf{Y}}, see Subsection 4.5, then TU=1subscript𝑇𝑈1T_{U}=1 in 𝐘/U𝐘𝑈{\mathbf{Y}}/U. In particular, when π𝜋\pi is unramified, T=1𝑇1T=1 and cTyp(λ)={1}cTyp𝜆1{\mathrm{cTyp}({\lambda})}=\{1\}.

When 𝐘𝐘{\mathbf{Y}} has enough points, it is possible to argue at stalks, and therefore the proposition follows from our version of Hilbert’s Theorem 90, Proposition 3.1.4(iii). The following argument applies even without the assumption of enough points.

Proof.

We must show that for all objects U𝑈U of 𝐘𝐘{\mathbf{Y}} and all rR(U)𝑟𝑅𝑈r\in R(U) satisfying rλr=1superscript𝑟𝜆𝑟1r^{\lambda}r=1, there is a covering VU𝑉𝑈V\to U and aR×(V)𝑎superscript𝑅𝑉a\in{R^{\times}}(V) such that a1aλ=rsuperscript𝑎1superscript𝑎𝜆𝑟a^{-1}a^{\lambda}=r in R(V)𝑅𝑉R(V).

Refining U𝑈U if necessary, we may assume that R(U)𝑅𝑈R(U) is a quadratic étale S(U)𝑆𝑈S(U)-algebra, see Subsection 3.2. By Proposition 3.1.4(iii), for all 𝔭SpecS(U)𝔭Spec𝑆𝑈{\mathfrak{p}}\in\operatorname{Spec}S(U), there is a𝔭R(U)𝔭×subscript𝑎𝔭𝑅superscriptsubscript𝑈𝔭a_{\mathfrak{p}}\in{R(U)_{\mathfrak{p}}^{\times}} such that a𝔭1a𝔭λ=rsuperscriptsubscript𝑎𝔭1superscriptsubscript𝑎𝔭𝜆𝑟a_{\mathfrak{p}}^{-1}a_{\mathfrak{p}}^{\lambda}=r. For each 𝔭𝔭{\mathfrak{p}}, choose some f𝔭S(U)𝔭subscript𝑓𝔭𝑆𝑈𝔭f_{\mathfrak{p}}\in S(U)-{\mathfrak{p}} such that a𝔭subscript𝑎𝔭a_{\mathfrak{p}} is the image of an element in R(U)f𝔭×𝑅superscriptsubscript𝑈subscript𝑓𝔭{R(U)_{f_{\mathfrak{p}}}^{\times}}, also denoted a𝔭subscript𝑎𝔭a_{\mathfrak{p}}, which satisfies a𝔭1a𝔭λ=rsuperscriptsubscript𝑎𝔭1superscriptsubscript𝑎𝔭𝜆𝑟a_{\mathfrak{p}}^{-1}a_{\mathfrak{p}}^{\lambda}=r in S(U)f𝔭𝑆subscript𝑈subscript𝑓𝔭S(U)_{f_{\mathfrak{p}}}. The set {f𝔭}𝔭subscriptsubscript𝑓𝔭𝔭\{f_{\mathfrak{p}}\}_{\mathfrak{p}} is not contained in any proper ideal of S(U)𝑆𝑈S(U) and therefore generates the unit ideal. Since S𝑆S is a local ring object, there exists a covering {U𝔭U}𝔭SpecS(U)subscriptsubscript𝑈𝔭𝑈𝔭Spec𝑆𝑈\{U_{\mathfrak{p}}\to U\}_{{\mathfrak{p}}\in\operatorname{Spec}S(U)} such that the image of f𝔭subscript𝑓𝔭f_{{\mathfrak{p}}} is invertible in S(U𝔭)𝑆subscript𝑈𝔭S(U_{\mathfrak{p}}) for all 𝔭𝔭{\mathfrak{p}}. Now take V=𝔭U𝔭𝑉subscriptsquare-union𝔭subscript𝑈𝔭V=\bigsqcup_{{\mathfrak{p}}}U_{\mathfrak{p}} and a𝑎a to be the image of (a𝔭)𝔭subscriptsubscript𝑎𝔭𝔭(a_{{\mathfrak{p}}})_{{\mathfrak{p}}} in R(V)=𝔭R(U𝔭)𝑅𝑉subscriptproduct𝔭𝑅subscript𝑈𝔭R(V)=\prod_{{\mathfrak{p}}}R(U_{\mathfrak{p}}). ∎

Proposition 5.2.3.

T𝑇T is a 222-torsion abelian sheaf.

Proof.

Let U𝑈U be an object of 𝐘𝐘{\mathbf{Y}} and tT(U)𝑡𝑇𝑈t\in T(U). By passing to a covering of U𝑈U, we may assume that t𝑡t is the image of some rN(U)𝑟𝑁𝑈r\in N(U). Since rλr=1superscript𝑟𝜆𝑟1r^{\lambda}r=1, we have r2=(r1)λ(r1)1superscript𝑟2superscriptsuperscript𝑟1𝜆superscriptsuperscript𝑟11r^{2}=(r^{-1})^{\lambda}(r^{-1})^{-1}, and thus t2=1superscript𝑡21t^{2}=1 in T(U)𝑇𝑈T(U). ∎

Let n=degA𝑛deg𝐴n=\operatorname{deg}A. Using Lemma 4.3.9, we shall freely identify the group PGLn(R)subscriptPGL𝑛𝑅\operatorname{PGL}_{n}(R) with 𝒜utR-alg(Mn×n(R))𝒜𝑢subscript𝑡R-algsubscriptM𝑛𝑛𝑅\mathcal{A}ut_{\textrm{$R$-alg}}(\mathrm{M}_{n\times n}(R)). We denote by

λtr𝜆tr-\lambda\text{\rm tr}

the automorphism of GLn(R)subscriptGL𝑛𝑅\operatorname{GL}_{n}(R) given by x(x1)λtrmaps-to𝑥superscriptsuperscript𝑥1𝜆trx\mapsto(x^{-1})^{\lambda\text{\rm tr}} on sections. This automorphism induces an automorphism on PGLn(R)subscriptPGL𝑛𝑅\operatorname{PGL}_{n}(R), which is also denoted λtr𝜆tr-\lambda\text{\rm tr}. We need the following lemma.

Lemma 5.2.4.

Let g𝑔g be a section of PGLn(R)=AutR-alg(Mn×n(R))=AutR-alg(ΛMn×n(R)op)subscriptPGL𝑛𝑅subscriptAutR-algsubscriptM𝑛𝑛𝑅subscriptAutR-algΛsubscriptM𝑛𝑛superscript𝑅op\operatorname{PGL}_{n}(R)=\operatorname{Aut}_{\text{\rm$R$-alg}}(\mathrm{M}_{n\times n}(R))=\operatorname{Aut}_{\text{\rm$R$-alg}}(\Lambda\mathrm{M}_{n\times n}(R)^{\text{op}}) and view λtr𝜆tr\lambda\text{\rm tr} as an R𝑅R-algebra isomorphism Mn×n(R)ΛMn×n(R)opsubscriptM𝑛𝑛𝑅ΛsubscriptM𝑛𝑛superscript𝑅op\mathrm{M}_{n\times n}(R)\to\Lambda\mathrm{M}_{n\times n}(R)^{\text{op}}. Then gλtr=λtrgλtr𝑔𝜆tr𝜆trsuperscript𝑔𝜆trg\circ\lambda\text{\rm tr}=\lambda\text{\rm tr}\circ g^{-\lambda\text{\rm tr}}.

Proof.

Suppose gPGLn(R)(U)𝑔subscriptPGL𝑛𝑅𝑈g\in\operatorname{PGL}_{n}(R)(U) for some object U𝑈U of 𝐗𝐗{\mathbf{X}}. It is enough to prove the equality after passing to a covering VU𝑉𝑈V\to U. We may therefore assume that g𝑔g is inner, and the lemma follows by computation. ∎

Construction 5.2.5.

Let (A,τ)𝐴𝜏(A,\tau) be a degree-n𝑛n Azumaya R𝑅R-algebra with a λ𝜆\lambda-involution. We now construct an element

ct(τ)cTyp(λ)=H0(T)ct𝜏cTyp𝜆superscriptH0𝑇\mathrm{ct}(\tau)\in{\mathrm{cTyp}({\lambda})}=\mathrm{H}^{0}(T)

and call it the coarse λ𝜆\lambda-type of τ𝜏\tau. This construction, which is concluded in Definition 5.2.7, will play a major role in the sequel.

Choose a covering U𝐘𝑈subscript𝐘U\to*_{\mathbf{Y}} such that there exists an isomorphism of RUsubscript𝑅𝑈R_{U}-algebras ψ:AUMn×n(RU):𝜓similar-tosubscript𝐴𝑈subscriptM𝑛𝑛subscript𝑅𝑈\psi:A_{U}\xrightarrow{\sim}\mathrm{M}_{n\times n}(R_{U}). The isomorphism ψ𝜓\psi gives rise to a λUsubscript𝜆𝑈\lambda_{U}-involution

σ=ψτUψ1:Mn×n(RU)Mn×n(RU).:𝜎𝜓subscript𝜏𝑈superscript𝜓1subscriptM𝑛𝑛subscript𝑅𝑈subscriptM𝑛𝑛subscript𝑅𝑈\sigma=\psi\circ\tau_{U}\circ\psi^{-1}:\mathrm{M}_{n\times n}(R_{U})\to\mathrm{M}_{n\times n}(R_{U})\ .

From σ𝜎\sigma and the involution λtr𝜆tr\lambda\text{\rm tr}, we construct

g:=λtrσAutRU-alg(Mn×n(RU))=PGLn(R)(U).assign𝑔𝜆tr𝜎subscriptAutRU-algsubscriptM𝑛𝑛subscript𝑅𝑈subscriptPGL𝑛𝑅𝑈g:=\lambda\text{\rm tr}\circ\sigma\in\operatorname{Aut}_{\text{$R_{U}$-alg}}(\mathrm{M}_{n\times n}(R_{U}))=\operatorname{PGL}_{n}(R)(U)\ .

By Lemma 5.2.4, we have id=σσ=λtrgλtrg=λtrλtrgλtrgid𝜎𝜎𝜆tr𝑔𝜆tr𝑔𝜆tr𝜆trsuperscript𝑔𝜆tr𝑔\mathrm{id}=\sigma\circ\sigma=\lambda\text{\rm tr}\circ g\circ\lambda\text{\rm tr}\circ g=\lambda\text{\rm tr}\circ\lambda\text{\rm tr}\circ g^{-\lambda\text{\rm tr}}\circ g, hence gλtrg=1superscript𝑔𝜆tr𝑔1g^{-\lambda\text{\rm tr}}g=1. Replacing U𝑈U by a covering UUsuperscript𝑈𝑈U^{\prime}\to U if necessary, we may assume that gPGLn(R)(U)𝑔subscriptPGL𝑛𝑅𝑈g\in\operatorname{PGL}_{n}(R)(U) lifts to a section

hGLn(R)(U).subscriptGL𝑛𝑅𝑈h\in\operatorname{GL}_{n}(R)(U)\ .

From gλtrg=1superscript𝑔𝜆tr𝑔1g^{-\lambda\text{\rm tr}}g=1, we get

(5) ε:=hλtrhR×(U).assign𝜀superscript𝜆trsuperscript𝑅𝑈\varepsilon:=h^{-\lambda\text{\rm tr}}h\in{R^{\times}}(U)\ .

Note that ελtrε=ελtrhλtrh=hλtrελtrh=hλtr(hλtrh1)h=1superscript𝜀𝜆tr𝜀superscript𝜀𝜆trsuperscript𝜆trsuperscript𝜆trsuperscript𝜀𝜆trsuperscript𝜆trsuperscript𝜆trsuperscript11\varepsilon^{\lambda\text{\rm tr}}\varepsilon=\varepsilon^{\lambda\text{\rm tr}}h^{-\lambda\text{\rm tr}}h=h^{-\lambda\text{\rm tr}}\varepsilon^{\lambda\text{\rm tr}}h=h^{-\lambda\text{\rm tr}}(h^{\lambda\text{\rm tr}}h^{-1})h=1, hence εN(U)𝜀𝑁𝑈\varepsilon\in N(U). Let ε¯¯𝜀\overline{\varepsilon} be the image of ε𝜀\varepsilon in T(U)𝑇𝑈T(U).

Lemma 5.2.6.

The section ε¯T(U)¯𝜀𝑇𝑈\overline{\varepsilon}\in T(U) determines a global section of T𝑇T. It is independent of the choices made in Construction 5.2.5.

Proof.

Let Usubscript𝑈U_{\bullet} denote the Čech hypercovering associated to U𝑈U—for the definition see Example 2.3.1. In particular, U0=Usubscript𝑈0𝑈U_{0}=U, U1=U×Usubscript𝑈1𝑈𝑈U_{1}=U\times U and d0,d1:U1U0:subscript𝑑0subscript𝑑1subscript𝑈1subscript𝑈0d_{0},d_{1}:U_{1}\to U_{0} are given by di(u0,u1)=u1isubscript𝑑𝑖subscript𝑢0subscript𝑢1subscript𝑢1𝑖d_{i}(u_{0},u_{1})=u_{1-i} on sections. Proving that ε¯¯𝜀\overline{\varepsilon} determines a global section of T𝑇T amounts to showing that there exists a covering VU1=U×U𝑉subscript𝑈1𝑈𝑈V\to U_{1}=U\times U and βR×(V)𝛽superscript𝑅𝑉\beta\in{R^{\times}}(V) such that d1ε1d0ε=β1βλsuperscriptsubscript𝑑1superscript𝜀1superscriptsubscript𝑑0𝜀superscript𝛽1superscript𝛽𝜆d_{1}^{*}\varepsilon^{-1}\cdot d_{0}^{*}\varepsilon=\beta^{-1}\beta^{\lambda} holds in R×(V)superscript𝑅𝑉{R^{\times}}(V).

For i{0,1}𝑖01i\in\{0,1\}, let ψisubscript𝜓𝑖\psi_{i} denote the pullback of ψ:AUMn×n(RU):𝜓subscript𝐴𝑈subscriptM𝑛𝑛subscript𝑅𝑈\psi:A_{U}\to\mathrm{M}_{n\times n}(R_{U}) along di:U1U0=U:subscript𝑑𝑖subscript𝑈1subscript𝑈0𝑈d_{i}:U_{1}\to U_{0}=U. Define σi:Mn×n(RU1)Mn×n(RU1):subscript𝜎𝑖subscriptM𝑛𝑛subscript𝑅subscript𝑈1subscriptM𝑛𝑛subscript𝑅subscript𝑈1\sigma_{i}:\mathrm{M}_{n\times n}(R_{U_{1}})\to\mathrm{M}_{n\times n}(R_{U_{1}}) similarly. Let

a:=ψ1ψ01:Mn×n(RU1)Mn×n(RU1).:assign𝑎subscript𝜓1superscriptsubscript𝜓01subscriptM𝑛𝑛subscript𝑅subscript𝑈1subscriptM𝑛𝑛subscript𝑅subscript𝑈1a:=\psi_{1}\circ\psi_{0}^{-1}:\mathrm{M}_{n\times n}(R_{U_{1}})\to\mathrm{M}_{n\times n}(R_{U_{1}}).

and regard a𝑎a as an element of PGLn(R)(U1)subscriptPGL𝑛𝑅subscript𝑈1\operatorname{PGL}_{n}(R)(U_{1}). The fact that ψi1σiψi=τU1superscriptsubscript𝜓𝑖1subscript𝜎𝑖subscript𝜓𝑖subscript𝜏subscript𝑈1\psi_{i}^{-1}\sigma_{i}\psi_{i}=\tau_{U_{1}} for i=0,1𝑖01i=0,1 implies that σ1a=aσ0subscript𝜎1𝑎𝑎subscript𝜎0\sigma_{1}\circ a=a\circ\sigma_{0}. Therefore, using Lemma 5.2.4, we get

d1gasuperscriptsubscript𝑑1𝑔𝑎\displaystyle d_{1}^{*}g\cdot a =λtrσ1a=λtraσ0=aλtrλtrσ0=aλtrd0g,absent𝜆trsubscript𝜎1𝑎𝜆tr𝑎subscript𝜎0superscript𝑎𝜆tr𝜆trsubscript𝜎0superscript𝑎𝜆trsuperscriptsubscript𝑑0𝑔\displaystyle=\lambda\text{\rm tr}\circ\sigma_{1}\circ a=\lambda\text{\rm tr}\circ a\circ\sigma_{0}=a^{-\lambda\text{\rm tr}}\circ\lambda\text{\rm tr}\circ\sigma_{0}=a^{-\lambda\text{\rm tr}}\cdot d_{0}^{*}g\ ,

or equivalently, aλtrd0ga1d1g1=1superscript𝑎𝜆trsuperscriptsubscript𝑑0𝑔superscript𝑎1superscriptsubscript𝑑1superscript𝑔11a^{-\lambda\text{\rm tr}}\cdot d_{0}^{*}g\cdot a^{-1}\cdot d_{1}^{*}g^{-1}=1 in PGLn(R)(U1)subscriptPGL𝑛𝑅subscript𝑈1\operatorname{PGL}_{n}(R)(U_{1}).

There exists a covering VU1𝑉subscript𝑈1V\to U_{1} such that the image of a𝑎a in PGLn(R)(V)subscriptPGL𝑛𝑅𝑉\operatorname{PGL}_{n}(R)(V), lifts to

bGLn(R)(V).𝑏subscriptGL𝑛𝑅𝑉b\in\operatorname{GL}_{n}(R)(V)\ .

The relation aλtrd0ga1d1g1=1superscript𝑎𝜆trsuperscriptsubscript𝑑0𝑔superscript𝑎1superscriptsubscript𝑑1superscript𝑔11a^{-\lambda\text{\rm tr}}\cdot d_{0}^{*}g\cdot a^{-1}\cdot d_{1}^{*}g^{-1}=1 now implies that

(6) β:=bλtrd0hb1d1h1R×(V).assign𝛽superscript𝑏𝜆trsuperscriptsubscript𝑑0superscript𝑏1superscriptsubscript𝑑1superscript1superscript𝑅𝑉\beta:=b^{-\lambda\text{\rm tr}}\cdot d_{0}^{*}h\cdot b^{-1}\cdot d_{1}^{*}h^{-1}\in{R^{\times}}(V)\ .

Using (5) and (6) we get

β1βλsuperscript𝛽1superscript𝛽𝜆\displaystyle\beta^{-1}\beta^{\lambda} =β1(bλtrd0hb1d1h1)λtrabsentsuperscript𝛽1superscriptsuperscript𝑏𝜆trsuperscriptsubscript𝑑0superscript𝑏1superscriptsubscript𝑑1superscript1𝜆tr\displaystyle=\beta^{-1}(b^{-\lambda\text{\rm tr}}\cdot d_{0}^{*}h\cdot b^{-1}\cdot d_{1}^{*}h^{-1})^{\lambda\text{\rm tr}}
=β1d1hλtrbλtrd0hλtrb1absentsuperscript𝛽1superscriptsubscript𝑑1superscript𝜆trsuperscript𝑏𝜆trsuperscriptsubscript𝑑0superscript𝜆trsuperscript𝑏1\displaystyle=\beta^{-1}\cdot d_{1}^{*}h^{-\lambda\text{\rm tr}}\cdot b^{-\lambda\text{\rm tr}}\cdot d_{0}^{*}h^{\lambda\text{\rm tr}}\cdot b^{-1}
=d1hλtr(bλtrd0hb1d1h1)1bλtrd0hλtrb1absentsuperscriptsubscript𝑑1superscript𝜆trsuperscriptsuperscript𝑏𝜆trsuperscriptsubscript𝑑0superscript𝑏1superscriptsubscript𝑑1superscript11superscript𝑏𝜆trsuperscriptsubscript𝑑0superscript𝜆trsuperscript𝑏1\displaystyle=d_{1}^{*}h^{-\lambda\text{\rm tr}}\cdot(b^{-\lambda\text{\rm tr}}\cdot d_{0}^{*}h\cdot b^{-1}\cdot d_{1}^{*}h^{-1})^{-1}\cdot b^{-\lambda\text{\rm tr}}\cdot d_{0}^{*}h^{\lambda\text{\rm tr}}\cdot b^{-1}
=d1hλtrd1hbd0h1bλtrbλtrd0hλtrb1=d1εd0ε1absentsuperscriptsubscript𝑑1superscript𝜆trsuperscriptsubscript𝑑1𝑏superscriptsubscript𝑑0superscript1superscript𝑏𝜆trsuperscript𝑏𝜆trsuperscriptsubscript𝑑0superscript𝜆trsuperscript𝑏1superscriptsubscript𝑑1𝜀superscriptsubscript𝑑0superscript𝜀1\displaystyle=d_{1}^{*}h^{-\lambda\text{\rm tr}}\cdot d_{1}^{*}h\cdot b\cdot d_{0}^{*}h^{-1}\cdot b^{\lambda\text{\rm tr}}\cdot b^{-\lambda\text{\rm tr}}\cdot d_{0}^{*}h^{\lambda\text{\rm tr}}\cdot b^{-1}=d_{1}^{*}\varepsilon\cdot d_{0}^{*}\varepsilon^{-1}

in GLn(R)(V)subscriptGL𝑛𝑅𝑉\operatorname{GL}_{n}(R)(V). This establishes the first part of the lemma.

Let t𝑡t denote the global section determined by ε¯¯𝜀\overline{\varepsilon}. The construction of t𝑡t involves choosing U𝑈U, ψ𝜓\psi and hGLn(U)subscriptGL𝑛𝑈h\in\operatorname{GL}_{n}(U) above. Suppose that tH(T)superscript𝑡H𝑇t^{\prime}\in\mathrm{H}(T) was obtained by replacing these choices with Usuperscript𝑈U^{\prime}, ψsuperscript𝜓\psi^{\prime} and hGLn(U)superscriptsubscriptGL𝑛superscript𝑈h^{\prime}\in\operatorname{GL}_{n}(U^{\prime}). We need to show that t=t𝑡superscript𝑡t=t^{\prime}.

Define g,σ,εsuperscript𝑔superscript𝜎superscript𝜀g^{\prime},\sigma^{\prime},\varepsilon^{\prime} as above using Usuperscript𝑈U^{\prime}, ψsuperscript𝜓\psi^{\prime}, hsuperscripth^{\prime} in place of U𝑈U, ψ𝜓\psi, hh. It is clear that refining the covering U𝑈U\to* does not affect t𝑡t. Therefore, refining U𝑈U\to* and Usuperscript𝑈U^{\prime}\to* to U×U𝑈superscript𝑈{U\times U^{\prime}}\to*, we may assume that U=U𝑈superscript𝑈U=U^{\prime}. Write ψ=uψsuperscript𝜓𝑢𝜓\psi^{\prime}=u\circ\psi with uAutRU(Mn×n(RU))=PGLn(R)(U)𝑢subscriptAutsubscript𝑅𝑈subscriptM𝑛𝑛subscript𝑅𝑈subscriptPGL𝑛𝑅𝑈u\in\operatorname{Aut}_{R_{U}}(\mathrm{M}_{n\times n}(R_{U}))=\operatorname{PGL}_{n}(R)(U). Then σ=ψτUψ1=uσu1superscript𝜎superscript𝜓subscript𝜏𝑈superscript𝜓1𝑢𝜎superscript𝑢1\sigma^{\prime}=\psi^{\prime}\circ\tau_{U}\circ\psi^{\prime-1}=u\circ\sigma\circ u^{-1}, and using Lemma 5.2.4, we get g=λtrσ=uλtrgu1superscript𝑔𝜆trsuperscript𝜎superscript𝑢𝜆tr𝑔superscript𝑢1g^{\prime}=\lambda\text{\rm tr}\circ\sigma^{\prime}=u^{-\lambda\text{\rm tr}}gu^{-1}. Refining U𝑈U\to* further, if necessary, we may assume that u𝑢u lifts to vGLn(R)(U)𝑣subscriptGL𝑛𝑅𝑈v\in\operatorname{GL}_{n}(R)(U). The relation g=uλtrgu1superscript𝑔superscript𝑢𝜆tr𝑔superscript𝑢1g^{\prime}=u^{-\lambda\text{\rm tr}}gu^{-1} implies that there is αRU×𝛼superscriptsubscript𝑅𝑈\alpha\in{R_{U}^{\times}} such that h=α1vλtrhv1superscriptsuperscript𝛼1superscript𝑣𝜆trsuperscript𝑣1h^{\prime}=\alpha^{-1}v^{-\lambda\text{\rm tr}}hv^{-1}. Thus,

εsuperscript𝜀\displaystyle\varepsilon^{\prime} =hλtrh=(α1vλtrhv1)λtrα1vλtrhv1=αλvhλtrvλtrα1vλtrhv1=αλα1εabsentsuperscript𝜆trsuperscriptsuperscriptsuperscript𝛼1superscript𝑣𝜆trsuperscript𝑣1𝜆trsuperscript𝛼1superscript𝑣𝜆trsuperscript𝑣1superscript𝛼𝜆𝑣superscript𝜆trsuperscript𝑣𝜆trsuperscript𝛼1superscript𝑣𝜆trsuperscript𝑣1superscript𝛼𝜆superscript𝛼1𝜀\displaystyle=h^{\prime-\lambda\text{\rm tr}}h^{\prime}=(\alpha^{-1}v^{-\lambda\text{\rm tr}}hv^{-1})^{-\lambda\text{\rm tr}}\alpha^{-1}v^{-\lambda\text{\rm tr}}hv^{-1}=\alpha^{\lambda}vh^{-\lambda\text{\rm tr}}v^{\lambda\text{\rm tr}}\alpha^{-1}v^{-\lambda\text{\rm tr}}hv^{-1}=\alpha^{\lambda}\alpha^{-1}\varepsilon

and ε¯=ε¯¯𝜀¯superscript𝜀\overline{\varepsilon}=\overline{\varepsilon^{\prime}} in T(U)𝑇𝑈T(U). This completes the proof. ∎

Definition 5.2.7.

Let (A,τ)𝐴𝜏(A,\tau) be an Azumaya R𝑅R-algebra with a λ𝜆\lambda-involution. The coarse λ𝜆\lambda-type or coarse type of τ𝜏\tau is the global section of T𝑇T determined by ε¯T(U)¯𝜀𝑇𝑈\overline{\varepsilon}\in T(U) constructed above. It shall be denoted ct(τ)ct𝜏\mathrm{ct}(\tau) or ct(A,τ)ct𝐴𝜏\mathrm{ct}(A,\tau).

Remark 5.2.8.

In accordance with Remark 5.1.2, we shall write the coarse type of a λ𝜆\lambda-involution τ𝜏\tau of an Azumaya 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}}-algebra, defined to be ct(πτ)ctsubscript𝜋𝜏\mathrm{ct}(\pi_{*}\tau), as ctπ(τ)subscriptct𝜋𝜏\mathrm{ct}_{\pi}(\tau).

Proposition 5.2.9.

Let (A,τ)𝐴𝜏(A,\tau), (A,τ)superscript𝐴superscript𝜏(A^{\prime},\tau^{\prime}) be Azumaya R𝑅R-algebras with λ𝜆\lambda-involutions. Then:

  1. (i)

    ct(A,τ)=ct(Mm×m(A),τtr)ct𝐴𝜏ctsubscriptM𝑚𝑚𝐴𝜏tr\mathrm{ct}(A,\tau)=\mathrm{ct}(\mathrm{M}_{m\times m}(A),\tau\text{\rm tr}) for all m𝑚m.

  2. (ii)

    ct(τRτ)=ct(τ)ct(τ)ctsubscripttensor-product𝑅𝜏superscript𝜏ct𝜏ctsuperscript𝜏\mathrm{ct}(\tau\otimes_{R}\tau^{\prime})=\mathrm{ct}(\tau)\cdot\mathrm{ct}(\tau^{\prime}) in H0(T)superscriptH0𝑇\mathrm{H}^{0}(T).

  3. (iii)

    If there is a covering V𝑉V\to* such that (AV,τV)(AV,τV)subscript𝐴𝑉subscript𝜏𝑉subscriptsuperscript𝐴𝑉subscriptsuperscript𝜏𝑉(A_{V},\tau_{V})\cong(A^{\prime}_{V},\tau^{\prime}_{V}), then ct(τ)=ct(τ)ct𝜏ctsuperscript𝜏\mathrm{ct}(\tau)=\mathrm{ct}(\tau^{\prime}).

Consequently, the map t(τ)ct(τ):Typ(λ)cTyp(λ):maps-tot𝜏ct𝜏Typ𝜆cTyp𝜆\mathrm{t}(\tau)\mapsto\mathrm{ct}(\tau):{\mathrm{Typ}({\lambda})}\to{\mathrm{cTyp}({\lambda})} is a well-defined morphism of monoids.

Proof.

Write n=degA𝑛deg𝐴n=\operatorname{deg}A and n=degAsuperscript𝑛degsuperscript𝐴n^{\prime}=\operatorname{deg}A^{\prime}. Define U,ψ,g,σ,h,ε𝑈𝜓𝑔𝜎𝜀U,\psi,g,\sigma,h,\varepsilon as in Construction 5.2.5, and analogously, define U,ψ,g,σ,h,εsuperscript𝑈superscript𝜓superscript𝑔superscript𝜎superscriptsuperscript𝜀U^{\prime},\psi^{\prime},g^{\prime},\sigma^{\prime},h^{\prime},\varepsilon^{\prime} using (A,τ)superscript𝐴superscript𝜏(A^{\prime},\tau^{\prime}) in place of (A,τ)𝐴𝜏(A,\tau).

  1. (i)

    The isomorphism ψ:AUMn×n(RU):𝜓subscript𝐴𝑈subscriptM𝑛𝑛subscript𝑅𝑈\psi:A_{U}\to\mathrm{M}_{n\times n}(R_{U}) gives rise to an isomorphism

    ψm:Mm×m(A)UMm×m(Mn×n(RU))=Mnm×nm(RU).:subscript𝜓𝑚subscriptM𝑚𝑚subscript𝐴𝑈subscriptM𝑚𝑚subscriptM𝑛𝑛subscript𝑅𝑈subscriptM𝑛𝑚𝑛𝑚subscript𝑅𝑈\psi_{m}:\mathrm{M}_{m\times m}(A)_{U}\to\mathrm{M}_{m\times m}(\mathrm{M}_{n\times n}(R_{U}))=\mathrm{M}_{nm\times nm}(R_{U}).

    Let σm=ψmλtrψm1subscript𝜎𝑚subscript𝜓𝑚𝜆trsuperscriptsubscript𝜓𝑚1\sigma_{m}=\psi_{m}\circ\lambda\text{\rm tr}\circ\psi_{m}^{-1}, let gm:=λtrσassignsubscript𝑔𝑚𝜆tr𝜎g_{m}:=\lambda\text{\rm tr}\circ\sigma and let hm=(hh)GLnm(R)(U)subscript𝑚direct-sumsubscriptGL𝑛𝑚𝑅𝑈h_{m}=(h\oplus\dots\oplus h)\in\operatorname{GL}_{nm}(R)(U). Straightforward computation shows that the image of hmsubscript𝑚h_{m} in PGLnm(R)(U)subscriptPGL𝑛𝑚𝑅𝑈\operatorname{PGL}_{nm}(R)(U) is gmsubscript𝑔𝑚g_{m}. This means that εm:=hmλtrhmassignsubscript𝜀𝑚superscriptsubscript𝑚𝜆trsubscript𝑚\varepsilon_{m}:=h_{m}^{-\lambda\text{\rm tr}}h_{m} coincides with ε=hλtrh𝜀superscript𝜆tr\varepsilon=h^{-\lambda\text{\rm tr}}h in N(U)𝑁𝑈N(U), and thus ct(A,τ)=ct(Mm×m(A),τtr)ct𝐴𝜏ctsubscriptM𝑚𝑚𝐴𝜏tr\mathrm{ct}(A,\tau)=\mathrm{ct}(\mathrm{M}_{m\times m}(A),\tau\text{\rm tr}).

  2. (ii)

    Consider ψ~=ψψ:AUAUMn×n(RU)Mn×n(RU)=Mnn×nn(RU):~𝜓tensor-product𝜓superscript𝜓tensor-productsubscript𝐴𝑈subscriptsuperscript𝐴𝑈tensor-productsubscriptM𝑛𝑛subscript𝑅𝑈subscriptMsuperscript𝑛superscript𝑛subscript𝑅𝑈subscriptM𝑛superscript𝑛𝑛superscript𝑛subscript𝑅𝑈\tilde{\psi}=\psi\otimes\psi^{\prime}:A_{U}\otimes A^{\prime}_{U}\to\mathrm{M}_{n\times n}(R_{U})\otimes\mathrm{M}_{n^{\prime}\times n^{\prime}}(R_{U})=\mathrm{M}_{nn^{\prime}\times nn^{\prime}}(R_{U}), let σ~=ψ~(ττ)ψ~1=σσ~𝜎~𝜓tensor-product𝜏superscript𝜏superscript~𝜓1tensor-product𝜎superscript𝜎\tilde{\sigma}=\tilde{\psi}\circ(\tau\otimes\tau^{\prime})\circ\tilde{\psi}^{-1}=\sigma\otimes\sigma^{\prime}, and let g~=λtrσ=gg~𝑔𝜆tr𝜎tensor-product𝑔superscript𝑔\tilde{g}=\lambda\text{\rm tr}\circ\sigma=g\otimes g^{\prime}. Define h~:=hhGLnn(R)(U)assign~tensor-productsuperscriptsubscriptGL𝑛superscript𝑛𝑅𝑈\tilde{h}:=h\otimes h^{\prime}\in\operatorname{GL}_{nn^{\prime}}(R)(U). Then h~~\tilde{h} maps onto g~~𝑔\tilde{g}, and we have h~λtrh~=(hλtrh)(hλtrh)superscript~𝜆tr~tensor-productsuperscript𝜆trsuperscript𝜆trsuperscript\tilde{h}^{-\lambda\text{\rm tr}}\tilde{h}=(h^{-\lambda\text{\rm tr}}h)\otimes(h^{\prime-\lambda\text{\rm tr}}h^{\prime}), which means ct(τRτ)=ct(τ)ct(τ)ctsubscripttensor-product𝑅𝜏superscript𝜏ct𝜏ctsuperscript𝜏\mathrm{ct}(\tau\otimes_{R}\tau^{\prime})=\mathrm{ct}(\tau)\cdot\mathrm{ct}(\tau^{\prime}).

  3. (iii)

    Fix an isomorphism η:(AV,τV)(AV,τV):𝜂subscriptsuperscript𝐴𝑉subscriptsuperscript𝜏𝑉subscript𝐴𝑉subscript𝜏𝑉\eta:(A^{\prime}_{V},\tau^{\prime}_{V})\to(A_{V},\tau_{V}) and, in the construction of ct(τ)ct𝜏\mathrm{ct}(\tau), choose a covering U𝑈U\to* factoring through V𝑉V\to*. Taking U=Usuperscript𝑈𝑈U^{\prime}=U and ψ:=ψηU:AUMn×n(R):assignsuperscript𝜓𝜓subscript𝜂𝑈subscriptsuperscript𝐴𝑈subscriptM𝑛𝑛𝑅\psi^{\prime}:=\psi\circ\eta_{U}:A^{\prime}_{U}\to\mathrm{M}_{n\times n}(R) in the construction of ct(A,τ)ctsuperscript𝐴superscript𝜏\mathrm{ct}(A^{\prime},\tau^{\prime}), we find that

    σ=ψτUψ1=ψηUτUηU1ψ1=ψτψ1=σ,superscript𝜎superscript𝜓subscriptsuperscript𝜏𝑈superscript𝜓1𝜓subscript𝜂𝑈subscriptsuperscript𝜏𝑈superscriptsubscript𝜂𝑈1superscript𝜓1𝜓𝜏superscript𝜓1𝜎\sigma^{\prime}=\psi^{\prime}\tau^{\prime}_{U}\psi^{\prime-1}=\psi\eta_{U}\tau^{\prime}_{U}\eta_{U}^{-1}\psi^{-1}=\psi\tau\psi^{-1}=\sigma\ ,

    so ct(τ)=ct(τ)ct𝜏ctsuperscript𝜏\mathrm{ct}(\tau)=\mathrm{ct}(\tau^{\prime}). ∎

Remark 5.2.10.

There are examples where Typ(λ)cTyp(λ)Typ𝜆cTyp𝜆{\mathrm{Typ}({\lambda})}\to{\mathrm{cTyp}({\lambda})} is not injective. For example, by Proposition 5.2.3, the image of Typ(λ)cTyp(λ)Typ𝜆cTyp𝜆{\mathrm{Typ}({\lambda})}\to{\mathrm{cTyp}({\lambda})} is a subgroup, so this map is not injective when Typ(λ)Typ𝜆{\mathrm{Typ}({\lambda})} is not a group, e.g. Example 5.1.4.

Definition 5.2.11.

An abelian group object G𝐺G of 𝐘𝐘{\mathbf{Y}} is said to have square roots locally if the the squaring map xx2:GG:maps-to𝑥superscript𝑥2𝐺𝐺x\mapsto x^{2}:G\to G is an epimorphism. That is, for any object U𝑈U of 𝐘𝐘{\mathbf{Y}} and xG(U)𝑥𝐺𝑈x\in G(U), there exists a covering VU𝑉𝑈V\to U and yG(V)𝑦𝐺𝑉y\in G(V) such that y2=xsuperscript𝑦2𝑥y^{2}=x.

Example 5.2.12.

If 𝐘𝐘\mathbf{Y} is the topos of a topological space with the ring sheaf of continuous functions into \mathbb{C} or the étale ringed topos of a scheme on which 222 is invertible, then the group 𝒪𝐘×superscriptsubscript𝒪𝐘{\mathcal{O}_{\mathbf{Y}}^{\times}} has square roots locally. Indeed, this holds at the stalks, because the stalks of 𝒪𝐘subscript𝒪𝐘{\mathcal{O}}_{\mathbf{Y}} are strictly henselian rings in which 222 is invertible—this is well known in the case of an étale ringed topos of a scheme, or proved in Appendix A in the case of a topological space. Furthermore, if 𝐘𝐘{\mathbf{Y}} is the fppf ringed topos of an arbitrary scheme Y𝑌Y, then 𝒪𝐘×superscriptsubscript𝒪𝐘{\mathcal{O}_{\mathbf{Y}}^{\times}} has square roots locally, because any U𝑈U-section has a square root over a degree-222 finite flat covering of U𝑈U.

The main result of this section is the following theorem, which shows that under mild assumptions, λ𝜆\lambda-involutions of Azumaya algebras of the same degree having the same coarse type are locally isomorphic. As a consequence, an analogue of the desired property (i) from Section 5.1 holds under the same assumptions. The theorem holds in particular when π:𝐗𝐘:𝜋𝐗𝐘\pi:{\mathbf{X}}\to{\mathbf{Y}} is induced by a good C2subscript𝐶2C_{2}-quotient of schemes on which 222 is invertible (see Example 4.3.3), or by a C2subscript𝐶2C_{2}-quotient of Hausdorff topological spaces (see Example 4.3.4).

Theorem 5.2.13.

Let 𝐗𝐗{\mathbf{X}} be a locally ringed topos with involution λ𝜆\lambda, let π:𝐗𝐘:𝜋𝐗𝐘\pi:{\mathbf{X}}\to{\mathbf{Y}} be an exact quotient relative to λ𝜆\lambda, and write R=π𝒪𝐗𝑅subscript𝜋subscript𝒪𝐗R=\pi_{*}{\mathcal{O}}_{\mathbf{X}} and S=𝒪𝐘𝑆subscript𝒪𝐘S={\mathcal{O}}_{\mathbf{Y}}. Suppose that S×superscript𝑆{S^{\times}} has square roots locally and at least one of the following conditions holds:

  1. (1)

    2S×2superscript𝑆2\in S^{\times}.

  2. (2)

    π:𝐗𝐘:𝜋𝐗𝐘\pi:\mathbf{X}\to\mathbf{Y} is unramified.

  3. (3)

    n𝑛n is odd.

Suppose (A,τ)𝐴𝜏(A,\tau) and (A,τ)superscript𝐴superscript𝜏(A^{\prime},\tau^{\prime}) are two degree-n𝑛n Azumaya R𝑅R-algebras with λ𝜆\lambda-involutions. Then the following are equivalent:

  1. (a)

    (A,τ)𝐴𝜏(A,\tau) and (A,τ)superscript𝐴superscript𝜏(A^{\prime},\tau^{\prime}) are locally isomorphic as R𝑅R-algebras with involution.

  2. (b)

    (A,τ)𝐴𝜏(A,\tau) and (A,τ)superscript𝐴superscript𝜏(A^{\prime},\tau^{\prime}) have the same type.

  3. (c)

    (A,τ)𝐴𝜏(A,\tau) and (A,τ)superscript𝐴superscript𝜏(A^{\prime},\tau^{\prime}) have the same coarse type.

Proof.

Statement (b) is implied by (a) by virtue of the definition of “type”, Definition 5.1.1. Then the implication of (c) by (b) is Proposition 5.2.9. The final implication, that of (a) by (c), is somewhat technical and it is given by Proposition 5.3.8 below. ∎

Corollary 5.2.14.

Suppose the assumptions of Theorem 5.2.13 hold and 2S×2superscript𝑆2\in S^{\times}. Then the map t(τ)ct(τ):Typ(λ)cTyp(λ):maps-tot𝜏ct𝜏Typ𝜆cTyp𝜆\mathrm{t}(\tau)\mapsto\mathrm{ct}(\tau):{\mathrm{Typ}({\lambda})}\to{\mathrm{cTyp}({\lambda})} is injective.

Proof.

Suppose ct(A,τ)=ct(A,τ)ct𝐴𝜏ctsuperscript𝐴superscript𝜏\mathrm{ct}(A,\tau)=\mathrm{ct}(A^{\prime},\tau^{\prime}) and write n=degA𝑛deg𝐴n=\operatorname{deg}A, n=degAsuperscript𝑛degsuperscript𝐴n^{\prime}=\operatorname{deg}A^{\prime}. By Proposition 5.2.9, we may replace (A,τ)𝐴𝜏(A,\tau) with (Mn×n(A),τtr)subscriptMsuperscript𝑛superscript𝑛𝐴𝜏tr(\mathrm{M}_{n^{\prime}\times n^{\prime}}(A),\tau\text{\rm tr}) and (A,τ)superscript𝐴superscript𝜏(A^{\prime},\tau^{\prime}) with (Mn×n(A),τtr)subscriptM𝑛𝑛superscript𝐴superscript𝜏tr(\mathrm{M}_{n\times n}(A^{\prime}),\tau^{\prime}\text{\rm tr}), and assume that degA=degAdeg𝐴degsuperscript𝐴\operatorname{deg}A=\operatorname{deg}A^{\prime}. Now, by Theorem 5.2.13, (A,τ)𝐴𝜏(A,\tau) is locally isomorphic to (A,τ)superscript𝐴superscript𝜏(A^{\prime},\tau^{\prime}) as an R𝑅R-algebra with involution, and a fortiori it has the same type. ∎

Corollary 5.2.15.

With the assumptions of Corollary 5.2.14, Typ(λ)Typ𝜆{\mathrm{Typ}({\lambda})} is a 222-torsion group.

Proof.

We know cTyp(λ)cTyp𝜆{\mathrm{cTyp}({\lambda})} is a 222-torsion group by Proposition 5.2.3, and Typ(λ)Typ𝜆{\mathrm{Typ}({\lambda})} is a submonoid by Corollary 5.2.14. ∎

Remark 5.2.16.

We do not know whether the assumptions of Corollary 5.2.14 imply that the map Typ(λ)cTyp(λ)Typ𝜆cTyp𝜆{\mathrm{Typ}({\lambda})}\to{\mathrm{cTyp}({\lambda})} is surjective. A more extensive discussion of this and some positive results will be given in Subsection 6.4.

The following theorem shows that the properties exhibited in Example 5.1.3 extend to our general setting under some assumptions.

Theorem 5.2.17.

With the assumptions of Theorem 5.2.13, the following hold:

  1. (i)

    If 2S×2superscript𝑆2\in{S^{\times}} and π:𝐗𝐘:𝜋𝐗𝐘\pi:{\mathbf{X}}\to{\mathbf{Y}} is a trivial quotient (Example 4.3.5), then Typ(λ)Typ𝜆{\mathrm{Typ}({\lambda})} is isomorphic to the group H0(μ2,R)superscriptH0subscript𝜇2𝑅\mathrm{H}^{0}(\mu_{2,R}).

  2. (ii)

    If π:𝐗𝐘:𝜋𝐗𝐘\pi:{\mathbf{X}}\to{\mathbf{Y}} is unramified, then Typ(λ)={1}Typ𝜆1{\mathrm{Typ}({\lambda})}=\{1\}.

  3. (iii)

    Let (A,τ)𝐴𝜏(A,\tau) be an Azumaya R𝑅R-algebra with a λ𝜆\lambda-involution. If degAdeg𝐴\operatorname{deg}A is odd, then t(τ)=1t𝜏1\mathrm{t}(\tau)=1.

We deduce Theorem 5.2.17 mostly as a corollary of Theorem 5.2.13.

Proof.

(i) This follows from Theorem 5.2.13 and Example 5.2.1 if we show that for every εH0(μ2,R)=H0(T)𝜀superscriptH0subscript𝜇2𝑅superscriptH0𝑇\varepsilon\in\mathrm{H}^{0}(\mu_{2,R})=\mathrm{H}^{0}(T), there is an involution of coarse type ε𝜀\varepsilon. To see that, let h=[0ε10]delimited-[]0𝜀10h=[\begin{smallmatrix}0&\varepsilon\\ 1&0\end{smallmatrix}] and take τ:M2×2(R)M2×2(R):𝜏subscriptM22𝑅subscriptM22𝑅\tau:\mathrm{M}_{2\times 2}(R)\to\mathrm{M}_{2\times 2}(R) defined by x(hxh1)λtrmaps-to𝑥superscript𝑥superscript1𝜆trx\mapsto(hxh^{-1})^{\lambda\text{\rm tr}}. That ct(τ)=εct𝜏𝜀\mathrm{ct}(\tau)=\varepsilon follows by applying Construction 5.2.5 with U=𝑈U=* and hh, ε𝜀\varepsilon just defined.

(ii) This follows from Theorem 5.2.13 and Proposition 5.2.2.

(iii) It is enough to show that ct(τ)=1ct𝜏1\mathrm{ct}(\tau)=1 whenever degA=2m+1deg𝐴2𝑚1\operatorname{deg}A=2m+1. Define U𝑈U, hh and ε𝜀\varepsilon as in Construction 5.2.5. From (5), we have h=εhλtr𝜀superscript𝜆trh=\varepsilon h^{\lambda\text{\rm tr}}. Taking the determinant of both sides yields deth=ε2m+1(deth)λsuperscript𝜀2𝑚1superscript𝜆\det h=\varepsilon^{2m+1}(\det h)^{\lambda}. Since ελ=ε1superscript𝜀𝜆superscript𝜀1\varepsilon^{\lambda}=\varepsilon^{-1}, this implies that ε=β1βλ𝜀superscript𝛽1superscript𝛽𝜆\varepsilon=\beta^{-1}\beta^{\lambda} for β=εmdeth𝛽superscript𝜀𝑚\beta=\varepsilon^{-m}\det h. This means that ε¯¯𝜀\overline{\varepsilon}, the image of ε𝜀\varepsilon in T(U)𝑇𝑈T(U), is trivial, so ct(τ)=1ct𝜏1\mathrm{ct}(\tau)=1. ∎

We note that part (i) applies in the case where π:𝐗𝐘:𝜋𝐗𝐘\pi:{\mathbf{X}}\to{\mathbf{Y}} is induced by a scheme X𝑋X on which 222 is invertible endowed with the trivial involution λ=id:XX:𝜆id𝑋𝑋\lambda=\mathrm{id}:X\to X; see Example 5.1.3(i).

Part (ii) applies to the case where π:𝐗𝐘:𝜋𝐗𝐘\pi:{\mathbf{X}}\to{\mathbf{Y}} is induced by a quadratic étale morphism of schemes π:XY:𝜋𝑋𝑌\pi:X\to Y where X𝑋X is given the canonical Y𝑌Y-involution; see Example 5.1.3(ii).

Our last application of Theorem 5.2.13 provides a concrete realization of the first cohomology set of the projective unitary group of an Azumaya R𝑅R-algebra with a λ𝜆\lambda-involution (A,τ)𝐴𝜏(A,\tau). As usual, the unitary group of (A,τ)𝐴𝜏(A,\tau) is the group object U(A,τ)U𝐴𝜏\operatorname{U}(A,\tau) in 𝐘𝐘{\mathbf{Y}} whose V𝑉V-sections are {aA(V):aτa=1}conditional-set𝑎𝐴𝑉superscript𝑎𝜏𝑎1\{a\in A(V)\,:\,a^{\tau}a=1\}, and the projective unitary group of (A,τ)𝐴𝜏(A,\tau) is the quotient

PU(A,τ)=U(A,τ)/NPU𝐴𝜏U𝐴𝜏𝑁\operatorname{PU}(A,\tau)=\operatorname{U}(A,\tau)/N

where N𝑁N, the group of λ𝜆\lambda-norm 111 elements in R𝑅R defined above.

If (A,τ)superscript𝐴superscript𝜏(A^{\prime},\tau^{\prime}) is another Azumaya R𝑅R-algebra with a λ𝜆\lambda-involution such that degA=degAdeg𝐴degsuperscript𝐴\operatorname{deg}A=\operatorname{deg}A^{\prime}, we further define omR((A,τ),(A,τ))𝑜subscript𝑚𝑅𝐴𝜏superscript𝐴superscript𝜏\mathcal{H}\!om_{R}((A,\tau),(A^{\prime},\tau^{\prime})) to be the sheaf of R𝑅R-linear isomorphisms from (A,τ)𝐴𝜏(A,\tau) to (A,τ)superscript𝐴superscript𝜏(A^{\prime},\tau^{\prime}), and 𝒜utR(A,τ)𝒜𝑢subscript𝑡𝑅𝐴𝜏\mathcal{A}ut_{R}(A,\tau) to be the group sheaf of R𝑅R-linear automorphisms of (A,τ)𝐴𝜏(A,\tau).

Lemma 5.2.18.

Suppose S×superscript𝑆{S^{\times}} has square roots locally. Let (A,τ)𝐴𝜏(A,\tau) be a degree-n𝑛n Azumaya R𝑅R-algebra with a λ𝜆\lambda-involution. Then the map U(A,τ)𝒜utR(A,τ)U𝐴𝜏𝒜𝑢subscript𝑡𝑅𝐴𝜏\operatorname{U}(A,\tau)\to\mathcal{A}ut_{R}(A,\tau) sending a section x𝑥x to conjugation by x𝑥x is an epimorphism with kernel N𝑁N. Consequently, it induces an isomorphism PU(A,τ)𝒜utR(A,τ)PU𝐴𝜏𝒜𝑢subscript𝑡𝑅𝐴𝜏\operatorname{PU}(A,\tau)\cong\mathcal{A}ut_{R}(A,\tau).

Proof.

That the kernel is N𝑁N follows easily from the fact that the centre of A𝐴A is R𝑅R. We turn to proving that the map is an epimorphism.

Let V𝐘𝑉𝐘V\in{\mathbf{Y}} and ψ𝒜utR(A,τ)(V)=AutRV(AV,τV)𝜓𝒜𝑢subscript𝑡𝑅𝐴𝜏𝑉subscriptAutsubscript𝑅𝑉subscript𝐴𝑉subscript𝜏𝑉\psi\in\mathcal{A}ut_{R}(A,\tau)(V)=\operatorname{Aut}_{R_{V}}(A_{V},\tau_{V}). By replacing V𝑉V with suitable covering, we may assume that AV=Mn×n(RV)subscript𝐴𝑉subscriptM𝑛𝑛subscript𝑅𝑉A_{V}=\mathrm{M}_{n\times n}(R_{V}) and that ψVPGLn(R)(V)subscript𝜓𝑉subscriptPGL𝑛𝑅𝑉\psi_{V}\in\operatorname{PGL}_{n}(R)(V) is given section-wise by ψV(x)=hxh1subscript𝜓𝑉𝑥𝑥superscript1\psi_{V}(x)=hxh^{-1} for some hGLn(R)(V)subscriptGL𝑛𝑅𝑉h\in\operatorname{GL}_{n}(R)(V). Since ψVτV=τVψVsubscript𝜓𝑉subscript𝜏𝑉subscript𝜏𝑉subscript𝜓𝑉\psi_{V}\circ\tau_{V}=\tau_{V}\circ\psi_{V}, for any section xA(V)𝑥𝐴𝑉x\in A(V), we have h1xτh=hτxτ(h1)τsuperscript1superscript𝑥𝜏superscript𝜏superscript𝑥𝜏superscriptsuperscript1𝜏h^{-1}x^{\tau}h=h^{\tau}x^{\tau}(h^{-1})^{\tau}, and thus hhτR×(V)superscript𝜏superscript𝑅𝑉hh^{\tau}\in{R^{\times}}(V). In fact, since (hhτ)λ=hhτsuperscriptsuperscript𝜏𝜆superscript𝜏(hh^{\tau})^{\lambda}=hh^{\tau}, we have hhτS×(V)superscript𝜏superscript𝑆𝑉hh^{\tau}\in{S^{\times}}(V). By assumption, we can replace V𝑉V with a suitable covering to assume that there is αS×(V)𝛼superscript𝑆𝑉\alpha\in{S^{\times}}(V) with α2=hhτsuperscript𝛼2superscript𝜏\alpha^{2}=hh^{\tau}. Replacing hh with hα1superscript𝛼1h\alpha^{-1} yields hhτ=1superscript𝜏1hh^{\tau}=1. We have therefore shown that over a covering of V𝑉V, ψ𝜓\psi lifts to a section of U(A,τ)U𝐴𝜏\operatorname{U}(A,\tau). ∎

Corollary 5.2.19.

With the assumptions of Theorem 5.2.13, let (A,τ)𝐴𝜏(A,\tau) be a degree-n𝑛n Azumaya R𝑅R-algebra with a λ𝜆\lambda-involution and identify 𝒜utR(A,τ)𝒜𝑢subscript𝑡𝑅𝐴𝜏\mathcal{A}ut_{R}(A,\tau) with PU(A,τ)PU𝐴𝜏\operatorname{PU}(A,\tau) as in Lemma 5.2.18. Then the functor (A,τ)omR((A,τ),(A,τ))maps-tosuperscript𝐴superscript𝜏𝑜subscript𝑚𝑅𝐴𝜏superscript𝐴superscript𝜏(A^{\prime},\tau^{\prime})\mapsto\mathcal{H}\!om_{R}((A,\tau),(A^{\prime},\tau^{\prime})) defines an equivalence between the full subcategory of Azn(𝐘,R,λ)subscriptAz𝑛𝐘𝑅𝜆{\text{\bf Az}}_{n}({\mathbf{Y}},R,\lambda) consisting of R𝑅R-algebras with a λ𝜆\lambda-involution of the same type as τ𝜏\tau and Tors(𝐘,PU(A,τ))Tors𝐘PU𝐴𝜏\text{\bf Tors}({\mathbf{Y}},\operatorname{PU}(A,\tau)). Consequently, H1(𝐘,PU(A,τ))superscriptH1𝐘PU𝐴𝜏\mathrm{H}^{1}({\mathbf{Y}},\operatorname{PU}(A,\tau)) is in canonical bijection with isomorphism classes of the aforementioned algebras with involution.

Proof.

Theorem 5.2.13 implies that an R𝑅R-algebra with a λ𝜆\lambda-involution (A,τ)superscript𝐴superscript𝜏(A^{\prime},\tau^{\prime}) is locally isomorphic to (A,τ)𝐴𝜏(A,\tau) if and only if A𝐴A is Azumaya of degree n𝑛n and τ𝜏\tau is of the same type as τsuperscript𝜏\tau^{\prime}. With this fact at hand, the equivalence is standard; see [giraud_cohomologie_1971, V.§4]. The last statement follows from Proposition 2.4.2(i). ∎

Many of the previous results require that 2S×2superscript𝑆2\in{S^{\times}}. Indeed, our arguments build on using the coarse type, which is too coarse if 222 is not invertible—see Remark 5.2.10. Nevertheless, we ask:

Question 5.2.20.

Does the equivalence between (a) and (b) in Theorem 5.2.13 hold without assuming any of the conditions (1), (2), (3)?

A particularly interesting case is the morphism of fppf ringed topoi associated to a finite locally free good C2subscript𝐶2C_{2}-quotient π:XY:𝜋𝑋𝑌\pi:X\to Y, where X𝑋X is a scheme on which 222 is not invertible (consult Remark 4.4.9).

5.3. Proof of Theorem 5.2.13

In this subsection we complete the proof of Theorem 5.2.13 by showing that condition (c) implies condition (a). This result is given as Proposition 5.3.8 below. The reader can skip this subsection without loss of continuity.

In the following lemmas, unless otherwise specified, A𝐴A will be a ring, λ:AA:𝜆𝐴𝐴\lambda:A\to A an involution, and B𝐵B will be the fixed ring of λ𝜆\lambda. We will write A¯¯𝐴\overline{A} for A/Jac(A)𝐴Jac𝐴A/\mathrm{Jac}(A), and λ¯:A¯A¯:¯𝜆¯𝐴¯𝐴\overline{\lambda}:\overline{A}\to\overline{A} for the involution induced on A¯¯𝐴\overline{A} by λ𝜆\lambda. Let ε{±1}𝜀plus-or-minus1\varepsilon\in\{\pm 1\} and let hGLn(A)subscriptGL𝑛𝐴h\in\operatorname{GL}_{n}(A) be an (ε,λtr)𝜀𝜆tr(\varepsilon,\lambda\text{\rm tr})-hermitian matrix, which is to say

h=εhλtr.𝜀superscript𝜆trh=\varepsilon h^{\lambda\text{\rm tr}}.

Let H:An×AnA:𝐻superscript𝐴𝑛superscript𝐴𝑛𝐴H:A^{n}\times A^{n}\to A be the (ε,λ)𝜀𝜆(\varepsilon,\lambda)-hermitian form associated to hh; it is given by h(x,y)=xλtrhy𝑥𝑦superscript𝑥𝜆tr𝑦h(x,y)=x^{\lambda\text{\rm tr}}hy where x,yAn𝑥𝑦superscript𝐴𝑛x,y\in A^{n} are written as column vectors. Let H¯¯𝐻\overline{H} denote the reduction of H𝐻H to A¯¯𝐴\overline{A}.

Lemma 5.3.1.

Assume B𝐵B is local. If λ¯idA¯¯𝜆subscriptid¯𝐴\overline{\lambda}\neq\mathrm{id}_{\overline{A}}, or ε1𝜀1\varepsilon\neq-1 in A¯¯𝐴\overline{A}, or n𝑛n is odd, then there exists vGLn(A)𝑣subscriptGL𝑛𝐴v\in\operatorname{GL}_{n}(A) such that vλtrhvsuperscript𝑣𝜆tr𝑣v^{\lambda\text{\rm tr}}hv is a diagonal matrix.

Proof.

Proving the lemma is equivalent to showing that H𝐻H is diagonalizable, i.e., has an orthogonal basis.

We first claim that H¯¯𝐻\overline{H} has an orthogonal basis. This is well known when A¯¯𝐴\overline{A} is a field; see [scharlau_quadratic_1985, Th. 7.6.3] for the case where λ¯id¯𝜆id\overline{\lambda}\neq\mathrm{id} or ε1𝜀1\varepsilon\neq-1 in A¯¯𝐴\overline{A}, and [albert_symmetric_1938, Thm. 6] for the case where λ¯=idA¯𝜆subscriptid𝐴\overline{\lambda}=\mathrm{id}_{A}, ε=1𝜀1\varepsilon=-1 in A¯¯𝐴\overline{A} and n𝑛n is odd. We note in passing that the second case can occur only when the characteristic of A¯¯𝐴\overline{A} is 222. If A¯¯𝐴\overline{A} is not a field, then Theorem 3.3.8 and Proposition 3.1.4(ii) imply that A¯k×k¯𝐴𝑘𝑘\overline{A}\cong k\times k, where k𝑘k is the residue field of B𝐵B, and λ¯¯𝜆\overline{\lambda} acts by interchanging the two copies of k𝑘k. In this case H¯¯𝐻\overline{H} is hyperbolic and the easy proof is left to the reader.

We now claim that any nondegenerate (ε,λ)𝜀𝜆(\varepsilon,\lambda)-hermitian form H𝐻H whose reduction H¯¯𝐻\overline{H} admits a diagonalization is diagonalizable, thus proving the lemma. Let {x¯1,,x¯n}A¯nsubscript¯𝑥1subscript¯𝑥𝑛superscript¯𝐴𝑛\{\overline{x}_{1},\dots,\overline{x}_{n}\}\subseteq\overline{A}^{n} be an orthogonal basis for H¯¯𝐻\overline{H} and let x1Ansubscript𝑥1superscript𝐴𝑛x_{1}\in A^{n} be an arbitrary lift of x¯1subscript¯𝑥1\overline{x}_{1}. Then H(x1,x1)A×𝐻subscript𝑥1subscript𝑥1superscript𝐴H(x_{1},x_{1})\in{A^{\times}} and hence An=x1Ax1superscript𝐴𝑛direct-sumsubscript𝑥1𝐴superscriptsubscript𝑥1perpendicular-toA^{n}=x_{1}A\oplus x_{1}^{\perp}. Write P=x1𝑃superscriptsubscript𝑥1perpendicular-toP=x_{1}^{\perp} and H1=H|P×Psubscript𝐻1evaluated-at𝐻𝑃𝑃H_{1}=H|_{P\times P}. The A𝐴A-module P𝑃P is free because A𝐴A is semilocal and P𝑃P is projective of constant A𝐴A-rank n1𝑛1n-1. Furthermore, since P¯=i=2nx¯iA¯¯𝑃superscriptsubscript𝑖2𝑛subscript¯𝑥𝑖¯𝐴\overline{P}=\sum_{i=2}^{n}\overline{x}_{i}\overline{A}, the form H1¯¯subscript𝐻1\overline{H_{1}} is diagonalizable by construction. We finish by applying induction to H1subscript𝐻1H_{1}. ∎

Lemma 5.3.2.

Assume B𝐵B is local, and suppose λ¯=idA¯¯𝜆subscriptid¯𝐴\overline{\lambda}=\mathrm{id}_{\overline{A}}, that ε=1𝜀1\varepsilon=-1 and that 2A×2superscript𝐴2\in{A^{\times}}. Then there exists vGLn(A)𝑣subscriptGL𝑛𝐴v\in\operatorname{GL}_{n}(A) such that vλtrhvsuperscript𝑣𝜆tr𝑣v^{\lambda\text{\rm tr}}hv is a direct sum of 2×2222\times 2 matrices in [0110]+M2×2(Jac(A))delimited-[]0110subscriptM22Jac𝐴[\begin{smallmatrix}0&1\\ -1&0\end{smallmatrix}]+\mathrm{M}_{2\times 2}(\mathrm{Jac}(A)). In particular, n𝑛n is even.

Proof.

We need to show that Ansuperscript𝐴𝑛A^{n} admits a basis {x1,y1,x2,y2,}subscript𝑥1subscript𝑦1subscript𝑥2subscript𝑦2\{x_{1},y_{1},x_{2},y_{2},\dots\} such that xiA+yiAsubscript𝑥𝑖𝐴subscript𝑦𝑖𝐴x_{i}A+y_{i}A is orthogonal to xjA+yjAsubscript𝑥𝑗𝐴subscript𝑦𝑗𝐴x_{j}A+y_{j}A whenever ij𝑖𝑗i\neq j and such that [H(xi,xi)H(xi,yi)H(yi,xi)H(yi,yi)][0110]+M2×2(Jac(A)))[\begin{smallmatrix}H(x_{i},x_{i})&H(x_{i},y_{i})\\ H(y_{i},x_{i})&H(y_{i},y_{i})\end{smallmatrix}]\in[\begin{smallmatrix}0&1\\ -1&0\end{smallmatrix}]+\mathrm{M}_{2\times 2}(\mathrm{Jac}(A))).

By Theorem 3.3.8, the assumption λ¯=idA¯¯𝜆subscriptid¯𝐴\overline{\lambda}=\mathrm{id}_{\overline{A}} implies that A𝐴A is local, hence A¯¯𝐴\overline{A} is a field of characteristic different from 222 and H¯¯𝐻\overline{H} is a nondegenerate alternating bilinear form. This means that n𝑛n must be even. Choose a nonzero x¯A¯n¯𝑥superscript¯𝐴𝑛\overline{x}\in\overline{A}^{n}. Since H¯¯𝐻\overline{H} is nondegenerate, there is y¯¯𝑦\overline{y} such that H¯(x¯,y¯)=1¯𝐻¯𝑥¯𝑦1\overline{H}(\overline{x},\overline{y})=1. Since H¯¯𝐻\overline{H} is alternating, we also have H¯(y¯,x¯)=1¯𝐻¯𝑦¯𝑥1\overline{H}(\overline{y},\overline{x})=-1 and H¯(x¯,x¯)=H¯(y¯,y¯)=0¯𝐻¯𝑥¯𝑥¯𝐻¯𝑦¯𝑦0\overline{H}(\overline{x},\overline{x})=\overline{H}(\overline{y},\overline{y})=0. Let x,yAn𝑥𝑦superscript𝐴𝑛x,y\in A^{n} be lifts of x¯¯𝑥\overline{x} and y¯¯𝑦\overline{y}. The previous equations imply that M:=[H(x,x)H(x,y)H(y,x)H(y,y)][0110]+M2×2(Jac(A))assign𝑀delimited-[]𝐻𝑥𝑥𝐻𝑥𝑦𝐻𝑦𝑥𝐻𝑦𝑦delimited-[]0110subscriptM22Jac𝐴M:=[\begin{smallmatrix}H(x,x)&H(x,y)\\ H(y,x)&H(y,y)\end{smallmatrix}]\in[\begin{smallmatrix}0&1\\ -1&0\end{smallmatrix}]+\mathrm{M}_{2\times 2}(\mathrm{Jac}(A)). In particular, M𝑀M is invertible, and so An=(xAyA){x,y}superscript𝐴𝑛direct-sumdirect-sum𝑥𝐴𝑦𝐴superscript𝑥𝑦perpendicular-toA^{n}=(xA\oplus yA)\oplus\{x,y\}^{\perp}. We proceed by induction on the restriction of H𝐻H to {x,y}superscript𝑥𝑦perpendicular-to\{x,y\}^{\perp}. ∎

Lemma 5.3.3.

Assume B𝐵B is local and 2B×2superscript𝐵2\in{B^{\times}}. Then Jac(A)2Jac(B)AJacsuperscript𝐴2Jac𝐵𝐴\mathrm{Jac}(A)^{2}\subseteq\mathrm{Jac}(B)A.

Proof.

Write 𝔪=Jac(B)𝔪Jac𝐵{\mathfrak{m}}=\mathrm{Jac}(B) and let x,yJac(A)𝑥𝑦Jac𝐴x,y\in\mathrm{Jac}(A). Then xλ+x,xλxJac(A)B𝔪superscript𝑥𝜆𝑥superscript𝑥𝜆𝑥Jac𝐴𝐵𝔪x^{\lambda}+x,x^{\lambda}x\in\mathrm{Jac}(A)\cap B\subseteq{\mathfrak{m}}. The equality x2(xλ+x)x+(xλx)=0superscript𝑥2superscript𝑥𝜆𝑥𝑥superscript𝑥𝜆𝑥0x^{2}-(x^{\lambda}+x)x+(x^{\lambda}x)=0 implies x2𝔪Asuperscript𝑥2𝔪𝐴x^{2}\in{\mathfrak{m}}A. Likewise, y2,(x+y)2𝔪Asuperscript𝑦2superscript𝑥𝑦2𝔪𝐴y^{2},(x+y)^{2}\in{\mathfrak{m}}A. We finish by noting that xy=12((x+y)2x2y2)𝑥𝑦12superscript𝑥𝑦2superscript𝑥2superscript𝑦2xy=\frac{1}{2}((x+y)^{2}-x^{2}-y^{2}). ∎

In the following lemmas, given a ring A𝐴A and aA𝑎𝐴a\in A, we write A[a]𝐴delimited-[]𝑎A[\sqrt{a}] to denote the ring A[T]/(T2a)𝐴delimited-[]𝑇superscript𝑇2𝑎A[T]/(T^{2}-a) and let a𝑎\sqrt{a} denote the image of T𝑇T in A[a]𝐴delimited-[]𝑎A[\sqrt{a}]. By induction, we define A[a1,a2,]=A[a1][a2,]A[T1,T2,]/(T12a1,T22a2,)𝐴subscript𝑎1subscript𝑎2𝐴delimited-[]subscript𝑎1subscript𝑎2𝐴subscript𝑇1subscript𝑇2superscriptsubscript𝑇12subscript𝑎1superscriptsubscript𝑇22subscript𝑎2A[\sqrt{a_{1}},\sqrt{a_{2}},\dots]=A[\sqrt{a_{1}}][\sqrt{a_{2}},\dots]\cong A[T_{1},T_{2},\dots]/(T_{1}^{2}-a_{1},T_{2}^{2}-a_{2},\dots). If λ:AA:𝜆𝐴𝐴\lambda:A\to A is an involution with fixed ring B𝐵B and aB𝑎𝐵a\in B, then λ𝜆\lambda extends to A[a]𝐴delimited-[]𝑎A[\sqrt{a}] by setting (a)λ=asuperscript𝑎𝜆𝑎(\sqrt{a})^{\lambda}=\sqrt{a}, and the fixed ring of λ:A[a]A[a]:𝜆𝐴delimited-[]𝑎𝐴delimited-[]𝑎\lambda:A[\sqrt{a}]\to A[\sqrt{a}] is B[a]𝐵delimited-[]𝑎B[\sqrt{a}].

Lemma 5.3.4.

Assume B𝐵B is local and suppose λ¯=idA¯¯𝜆subscriptid¯𝐴\overline{\lambda}=\mathrm{id}_{\overline{A}}, that ε=1𝜀1\varepsilon=-1 and 2A×2superscript𝐴2\in{A^{\times}}. Suppose that hh lies in [0110]+M2×2(Jac(A))delimited-[]0110subscriptM22Jac𝐴[\begin{smallmatrix}0&1\\ -1&0\end{smallmatrix}]+\mathrm{M}_{2\times 2}(\mathrm{Jac}(A)). Then there are s,tB×𝑠𝑡superscript𝐵s,t\in{B^{\times}}, f1,,frB[s,t]subscript𝑓1subscript𝑓𝑟𝐵𝑠𝑡f_{1},\dots,f_{r}\in{B[\sqrt{s},\sqrt{t}]} and viGL2(A[s,t]fi)subscript𝑣𝑖subscriptGL2𝐴subscript𝑠𝑡subscript𝑓𝑖v_{i}\in\operatorname{GL}_{2}(A[\sqrt{s},\sqrt{t}]_{f_{i}}) (i=1,,r𝑖1𝑟i=1,\dots,r) such that ifiB[s,t]=B[s,t]subscript𝑖subscript𝑓𝑖𝐵𝑠𝑡𝐵𝑠𝑡\sum_{i}f_{i}B[\sqrt{s},\sqrt{t}]=B[\sqrt{s},\sqrt{t}] and viλtrhvi=[0110]superscriptsubscript𝑣𝑖𝜆trsubscript𝑣𝑖delimited-[]0110v_{i}^{\lambda\text{\rm tr}}hv_{i}=[\begin{smallmatrix}0&1\\ -1&0\end{smallmatrix}] in A[s,t]fi𝐴subscript𝑠𝑡subscript𝑓𝑖A[\sqrt{s},\sqrt{t}]_{f_{i}} for all i𝑖i.

Proof.

Again, by Theorem 3.3.8, A𝐴A is local. We write 𝔪=Jac(B)𝔪Jac𝐵{\mathfrak{m}}=\mathrm{Jac}(B), 𝔐=Jac(A)𝔐Jac𝐴{\mathfrak{M}}=\mathrm{Jac}(A) and let {x,y}𝑥𝑦\{x,y\} denote the standard A𝐴A-basis of A2superscript𝐴2A^{2}. Once s,tS𝑠𝑡𝑆s,t\in S and f1,,frB[s,t]subscript𝑓1subscript𝑓𝑟𝐵𝑠𝑡f_{1},\dots,f_{r}\in B[\sqrt{s},\sqrt{t}] above have been chosen, we need to show that for all i𝑖i, there are x~,y~A[s,t]fi2~𝑥~𝑦𝐴superscriptsubscript𝑠𝑡subscript𝑓𝑖2\tilde{x},\tilde{y}\in A[\sqrt{s},\sqrt{t}]_{f_{i}}^{2} such that [H(x~,x~)H(x~,y~)H(y~,x~)H(y~,y~)]=[0110]delimited-[]𝐻~𝑥~𝑥𝐻~𝑥~𝑦𝐻~𝑦~𝑥𝐻~𝑦~𝑦delimited-[]0110\big{[}\begin{smallmatrix}H(\tilde{x},\tilde{x})&H(\tilde{x},\tilde{y})\\ H(\tilde{y},\tilde{x})&H(\tilde{y},\tilde{y})\end{smallmatrix}\big{]}=[\begin{smallmatrix}0&1\\ -1&0\end{smallmatrix}].

Our assumption on hh implies that H(x,y)1+Jac(A)A×𝐻𝑥𝑦1Jac𝐴superscript𝐴H(x,y)\in 1+\mathrm{Jac}(A)\subseteq{A^{\times}}. Replacing y𝑦y with H(x,y)1y𝐻superscript𝑥𝑦1𝑦H(x,y)^{-1}y, we may assume that H(x,y)=1𝐻𝑥𝑦1H(x,y)=1 and so H(y,x)=1𝐻𝑦𝑥1H(y,x)=-1. We further write α=H(x,x)𝛼𝐻𝑥𝑥\alpha=H(x,x) and β=H(y,y)𝛽𝐻𝑦𝑦\beta=H(y,y). Since h=hλtrsuperscript𝜆trh=-h^{\lambda\text{\rm tr}}, we have αλ=αsuperscript𝛼𝜆𝛼\alpha^{\lambda}=-\alpha and βλ=βsuperscript𝛽𝜆𝛽\beta^{\lambda}=-\beta, and since λ¯=idA¯¯𝜆subscriptid¯𝐴\overline{\lambda}=\mathrm{id}_{\overline{A}}, it follows that α,β𝔐𝛼𝛽𝔐\alpha,\beta\in{\mathfrak{M}} and hence α2,β2B𝔐𝔪superscript𝛼2superscript𝛽2𝐵𝔐𝔪\alpha^{2},\beta^{2}\in B\cap{\mathfrak{M}}\subseteq{\mathfrak{m}}.

Observe that the polynomial p(x)=2x2+(22α)x+αA[x]𝑝𝑥2superscript𝑥222𝛼𝑥𝛼𝐴delimited-[]𝑥p(x)=-2x^{2}+(2-2\alpha)x+\alpha\in A[x] has discriminant

s:=(22α)24(2α)=4+4α2assign𝑠superscript22𝛼242𝛼44superscript𝛼2s:=(2-2\alpha)^{2}-4(-2\alpha)=4+4\alpha^{2}

and that s4+𝔪B×𝑠4𝔪superscript𝐵s\in 4+{\mathfrak{m}}\subseteq{B^{\times}}. The roots of p(x)𝑝𝑥p(x) in A[s]𝐴delimited-[]𝑠A[\sqrt{s}] are c1=14(s+22α)subscript𝑐114𝑠22𝛼c_{1}=\frac{1}{4}(\sqrt{s}+2-2\alpha) and c2=14(s+22α)subscript𝑐214𝑠22𝛼c_{2}=\frac{1}{4}(-\sqrt{s}+2-2\alpha) and we note that

(7) 16(c1λc1+c2λc2)16superscriptsubscript𝑐1𝜆subscript𝑐1superscriptsubscript𝑐2𝜆subscript𝑐2\displaystyle 16(c_{1}^{\lambda}c_{1}+c_{2}^{\lambda}c_{2}) =(s+2)24α2+(s2)24α2=16B×.absentsuperscript𝑠224superscript𝛼2superscript𝑠224superscript𝛼216superscript𝐵\displaystyle=(\sqrt{s}+2)^{2}-4\alpha^{2}+(\sqrt{s}-2)^{2}-4\alpha^{2}=16\in{B^{\times}}\ .

We repeat this construction with β𝛽\beta in place of α𝛼\alpha, denoting the elements corresponding to s𝑠s, c1subscript𝑐1c_{1}, c2subscript𝑐2c_{2} by t𝑡t, d1subscript𝑑1d_{1}, d2subscript𝑑2d_{2}.

Fix a maximal ideal 𝔭B[s,t]subgroup-of𝔭𝐵𝑠𝑡{\mathfrak{p}}\lhd B[\sqrt{s},\sqrt{t}] and write B:=B[s,t]𝔭assignsuperscript𝐵𝐵subscript𝑠𝑡𝔭B^{\prime}:=B[\sqrt{s},\sqrt{t}]_{\mathfrak{p}}, A:=A[s,t]𝔭assignsuperscript𝐴𝐴subscript𝑠𝑡𝔭A^{\prime}:=A[\sqrt{s},\sqrt{t}]_{\mathfrak{p}}, 𝔪=Jac(B)superscript𝔪Jacsuperscript𝐵{\mathfrak{m}}^{\prime}=\mathrm{Jac}(B^{\prime}) and 𝔐=Jac(A)superscript𝔐Jacsuperscript𝐴{\mathfrak{M}}^{\prime}=\mathrm{Jac}(A^{\prime}).

It is clear that Bsuperscript𝐵B^{\prime} is local. We claim that Asuperscript𝐴A^{\prime} is also local and 𝔐A𝔐𝔐superscript𝐴superscript𝔐{\mathfrak{M}}A^{\prime}\subseteq{\mathfrak{M}}^{\prime}. Indeed, by [reiner_maximal_1975-1, Th. 6.15], we have 𝔪B[s,t]Jac(B[s,t])𝔭𝔪𝐵𝑠𝑡Jac𝐵𝑠𝑡𝔭{\mathfrak{m}}B[\sqrt{s},\sqrt{t}]\subseteq\mathrm{Jac}(B[\sqrt{s},\sqrt{t}])\subseteq{\mathfrak{p}}, and hence 𝔪B𝔭𝔭=𝔪𝔪superscript𝐵subscript𝔭𝔭superscript𝔪{\mathfrak{m}}B^{\prime}\subseteq{\mathfrak{p}}_{\mathfrak{p}}={\mathfrak{m}}^{\prime}, which in turn implies 𝔪A𝔪A𝔪superscript𝐴superscript𝔪superscript𝐴{\mathfrak{m}}A^{\prime}\subseteq{\mathfrak{m}}^{\prime}A^{\prime}. By Lemma 3.3.10, we have 𝔪A𝔐superscript𝔪superscript𝐴superscript𝔐{\mathfrak{m}}^{\prime}A^{\prime}\subseteq{\mathfrak{M}}^{\prime}, and by Lemma 5.3.3, 𝔐2𝔪Asuperscript𝔐2𝔪𝐴{\mathfrak{M}}^{2}\subseteq{\mathfrak{m}}A. Using the last three inclusions, we get (𝔐A)2=𝔐2A𝔪A𝔪A𝔐superscript𝔐superscript𝐴2superscript𝔐2superscript𝐴𝔪superscript𝐴superscript𝔪superscript𝐴superscript𝔐({\mathfrak{M}}A^{\prime})^{2}={\mathfrak{M}}^{2}A^{\prime}\subseteq{\mathfrak{m}}A^{\prime}\subseteq{\mathfrak{m}}^{\prime}A^{\prime}\subseteq{\mathfrak{M}}^{\prime}, and since 𝔐superscript𝔐{\mathfrak{M}}^{\prime} is semiprime, 𝔐A𝔐𝔐superscript𝐴superscript𝔐{\mathfrak{M}}A^{\prime}\subseteq{\mathfrak{M}}^{\prime}. The latter implies that AA/𝔐superscript𝐴superscript𝐴superscript𝔐A^{\prime}\to A^{\prime}/{\mathfrak{M}}^{\prime} factors through A/𝔐AA¯BBsuperscript𝐴𝔐superscript𝐴subscripttensor-product𝐵¯𝐴superscript𝐵A^{\prime}/{\mathfrak{M}}A^{\prime}\cong\overline{A}\otimes_{B}B^{\prime}, and hence the specialization of λ𝜆\lambda to A/𝔐superscript𝐴superscript𝔐A^{\prime}/{\mathfrak{M}}^{\prime} is the identity. Since Bsuperscript𝐵B^{\prime} is flat over B𝐵B, the local ring Bsuperscript𝐵B^{\prime} is the fixed ring of λ:AA:𝜆superscript𝐴superscript𝐴\lambda:A^{\prime}\to A^{\prime} and Theorem 3.3.8 implies that Asuperscript𝐴A^{\prime} is local.

Now, the inclusion 𝔐A𝔐𝔐superscript𝐴superscript𝔐{\mathfrak{M}}A^{\prime}\subseteq{\mathfrak{M}}^{\prime} implies α,β𝔐𝛼𝛽superscript𝔐\alpha,\beta\in{\mathfrak{M}}^{\prime}. By equation (7), there is i{1,2}𝑖12i\in\{1,2\} such that ciλciB×superscriptsubscript𝑐𝑖𝜆subscript𝑐𝑖superscript𝐵c_{i}^{\lambda}c_{i}\in{B^{\prime\times}}, and hence ciA×subscript𝑐𝑖superscript𝐴c_{i}\in{A^{\prime\times}}. In the same way, there is j{1,2}𝑗12j\in\{1,2\} such that djA×subscript𝑑𝑗superscript𝐴d_{j}\in{A^{\prime\times}}.

Working in Asuperscript𝐴A^{\prime}, we have

(8) ci1+(ci1)λsuperscriptsubscript𝑐𝑖1superscriptsuperscriptsubscript𝑐𝑖1𝜆\displaystyle c_{i}^{-1}+(c_{i}^{-1})^{\lambda} =4±s+22α+4±s+2+2αabsent4plus-or-minus𝑠22𝛼4plus-or-minus𝑠22𝛼\displaystyle=\frac{4}{\pm\sqrt{s}+2-2\alpha}+\frac{4}{\pm\sqrt{s}+2+2\alpha}
=8(±s+2)(±s+2)24α2absent8plus-or-minus𝑠2superscriptplus-or-minus𝑠224superscript𝛼2\displaystyle=\frac{8(\pm\sqrt{s}+2)}{(\pm\sqrt{s}+2)^{2}-4\alpha^{2}}
=8(±s+2)s±4s+44α2absent8plus-or-minus𝑠2plus-or-minus𝑠4𝑠44superscript𝛼2\displaystyle=\frac{8(\pm\sqrt{s}+2)}{s\pm 4\sqrt{s}+4-4\alpha^{2}}
=8(±s+2)±4s+8=2absent8plus-or-minus𝑠2plus-or-minus4𝑠82\displaystyle=\frac{8(\pm\sqrt{s}+2)}{\pm 4\sqrt{s}+8}=2

and likewise for djsubscript𝑑𝑗d_{j}. Write u=ci11𝑢superscriptsubscript𝑐𝑖11u=c_{i}^{-1}-1. Since α(ci1)2+(22α)ci12=ci2p(ci)=0𝛼superscriptsuperscriptsubscript𝑐𝑖1222𝛼superscriptsubscript𝑐𝑖12superscriptsubscript𝑐𝑖2𝑝subscript𝑐𝑖0\alpha(c_{i}^{-1})^{2}+(2-2\alpha)c_{i}^{-1}-2=c_{i}^{-2}p(c_{i})=0, we have

(9) αu2+2uα=0,𝛼superscript𝑢22𝑢𝛼0\alpha u^{2}+2u-\alpha=0\ ,

and since α𝔐𝛼superscript𝔐\alpha\in{\mathfrak{M}}^{\prime}, this implies u𝔐𝑢superscript𝔐u\in{\mathfrak{M}}^{\prime}. We further have uλ=usuperscript𝑢𝜆𝑢u^{\lambda}=-u by (8). In the same way, writing w=dj11𝑤superscriptsubscript𝑑𝑗11w=d_{j}^{-1}-1, we have βw2+2wβ=0𝛽superscript𝑤22𝑤𝛽0\beta w^{2}+2w-\beta=0, w𝔐𝑤superscript𝔐w\in{\mathfrak{M}}^{\prime} and wλ=wsuperscript𝑤𝜆𝑤w^{\lambda}=-w. Let x~=(1+u)x+(1w)y~𝑥1𝑢𝑥1𝑤𝑦\tilde{x}=(1+u)x+(1-w)y. Then, using (9), we get

H(x~,x~)𝐻~𝑥~𝑥\displaystyle H(\tilde{x},\tilde{x}) =(1+u)λ(1+u)α+(1+u)λ(1w)(1w)λ(1+u)+(1w)λ(1w)βabsentsuperscript1𝑢𝜆1𝑢𝛼superscript1𝑢𝜆1𝑤superscript1𝑤𝜆1𝑢superscript1𝑤𝜆1𝑤𝛽\displaystyle=(1+u)^{\lambda}(1+u)\alpha+(1+u)^{\lambda}(1-w)-(1-w)^{\lambda}(1+u)+(1-w)^{\lambda}(1-w)\beta
=(1u2)α+(1u)(1w)(1+w)(1+u)+(1w2)βabsent1superscript𝑢2𝛼1𝑢1𝑤1𝑤1𝑢1superscript𝑤2𝛽\displaystyle=(1-u^{2})\alpha+(1-u)(1-w)-(1+w)(1+u)+(1-w^{2})\beta
=ααu22u+ββw22w=0absent𝛼𝛼superscript𝑢22𝑢𝛽𝛽superscript𝑤22𝑤0\displaystyle=\alpha-\alpha u^{2}-2u+\beta-\beta w^{2}-2w=0

Likewise, y~:=(1u)x(1+w)yassign~𝑦1𝑢𝑥1𝑤𝑦\tilde{y}:=(1-u)x-(1+w)y satisfies H(y~,y~)=0𝐻~𝑦~𝑦0H(\tilde{y},\tilde{y})=0. Since u,w𝔐=Jac(A)𝑢𝑤superscript𝔐Jacsuperscript𝐴u,w\in{\mathfrak{M}}^{\prime}=\mathrm{Jac}(A^{\prime}), the vectors x~,y~~𝑥~𝑦\tilde{x},\tilde{y} span A2superscript𝐴2A^{\prime 2}. Since H𝐻H is nondegenerate, this forces H(x~,y~)A×𝐻~𝑥~𝑦superscript𝐴H(\tilde{x},\tilde{y})\in{A^{\prime\times}}. Replacing y~~𝑦\tilde{y} with H(x~,y~)1y~𝐻superscript~𝑥~𝑦1~𝑦H(\tilde{x},\tilde{y})^{-1}\tilde{y}, we find that [H(x~,x~)H(x~,y~)H(y~,x~)H(y~,y~)]=[0110]delimited-[]𝐻~𝑥~𝑥𝐻~𝑥~𝑦𝐻~𝑦~𝑥𝐻~𝑦~𝑦delimited-[]0110\big{[}\begin{smallmatrix}H(\tilde{x},\tilde{x})&H(\tilde{x},\tilde{y})\\ H(\tilde{y},\tilde{x})&H(\tilde{y},\tilde{y})\end{smallmatrix}\big{]}=[\begin{smallmatrix}0&1\\ -1&0\end{smallmatrix}].

Finally, for every maximal ideal 𝔭B[s,t]subgroup-of𝔭𝐵𝑠𝑡{\mathfrak{p}}\lhd B[\sqrt{s},\sqrt{t}], choose f𝔭B[t,s]𝔭subscript𝑓𝔭𝐵𝑡𝑠𝔭f_{\mathfrak{p}}\in B[\sqrt{t},\sqrt{s}]-{\mathfrak{p}} such that the coefficients of x~~𝑥\tilde{x}, y~~𝑦\tilde{y} constructed above are defined in B[t,s]f𝔭𝐵subscript𝑡𝑠subscript𝑓𝔭B[\sqrt{t},\sqrt{s}]_{f_{\mathfrak{p}}} and such that the identity [H(x~,x~)H(x~,y~)H(y~,x~)H(y~,y~)]=[0110]delimited-[]𝐻~𝑥~𝑥𝐻~𝑥~𝑦𝐻~𝑦~𝑥𝐻~𝑦~𝑦delimited-[]0110\big{[}\begin{smallmatrix}H(\tilde{x},\tilde{x})&H(\tilde{x},\tilde{y})\\ H(\tilde{y},\tilde{x})&H(\tilde{y},\tilde{y})\end{smallmatrix}\big{]}=[\begin{smallmatrix}0&1\\ -1&0\end{smallmatrix}] holds in B[s,t]f𝔭𝐵subscript𝑠𝑡subscript𝑓𝔭B[\sqrt{s},\sqrt{t}]_{f_{\mathfrak{p}}}. Then 𝔭f𝔭B[s,t]=B[s,t]subscript𝔭subscript𝑓𝔭𝐵𝑠𝑡𝐵𝑠𝑡\sum_{\mathfrak{p}}f_{\mathfrak{p}}B[\sqrt{s},\sqrt{t}]=B[\sqrt{s},\sqrt{t}]. Since B[s,t]𝐵𝑠𝑡B[\sqrt{s},\sqrt{t}] is a finite algebra over a local ring, it has only finitely many maximal ideals. The result follows. ∎

Lemma 5.3.5.

Maintaining the assumptions made at the beginning of this subsection, suppose that hGLn(A)superscriptsubscriptGL𝑛𝐴h^{\prime}\in\operatorname{GL}_{n}(A) is another (ε,λtr)𝜀𝜆tr(\varepsilon,\lambda\text{\rm tr})-hermitian matrix. Suppose further we are given a prime ideal 𝔭SpecB𝔭Spec𝐵{\mathfrak{p}}\in\operatorname{Spec}B, units s1,,sB𝔭×subscript𝑠1subscript𝑠superscriptsubscript𝐵𝔭s_{1},\dots,s_{\ell}\in{B_{\mathfrak{p}}^{\times}}, elements f1,,frB𝔭[s1,,s]subscript𝑓1subscript𝑓𝑟subscript𝐵𝔭subscript𝑠1subscript𝑠f_{1},\dots,f_{r}\in B_{\mathfrak{p}}[\sqrt{s_{1}},\dots,\sqrt{s_{\ell}}] and viGLn(A𝔭[s1,,s]fi)subscript𝑣𝑖subscriptGL𝑛subscript𝐴𝔭subscriptsubscript𝑠1subscript𝑠subscript𝑓𝑖v_{i}\in\operatorname{GL}_{n}(A_{\mathfrak{p}}[\sqrt{s_{1}},\dots,\sqrt{s_{\ell}}]_{f_{i}}) (i=1,,r𝑖1𝑟i=1,\dots,r) such that f1,,frsubscript𝑓1subscript𝑓𝑟f_{1},\dots,f_{r} generate the unit ideal in B𝔭[s1,,s]subscript𝐵𝔭subscript𝑠1subscript𝑠B_{\mathfrak{p}}[\sqrt{s_{1}},\dots,\sqrt{s_{\ell}}] and such that viλtrhvi=hsuperscriptsubscript𝑣𝑖𝜆trsubscript𝑣𝑖superscriptv_{i}^{\lambda\text{\rm tr}}hv_{i}=h^{\prime} for all 1ir1𝑖𝑟1\leq i\leq r. Then there is bB𝔭𝑏𝐵𝔭b\in B-{\mathfrak{p}} for which the previous condition holds upon replacing B𝔭subscript𝐵𝔭B_{\mathfrak{p}}, A𝔭subscript𝐴𝔭A_{\mathfrak{p}} with Bbsubscript𝐵𝑏B_{b}, Absubscript𝐴𝑏A_{b}.

Proof.

There is bB𝑏𝐵b\in B such that s1,,ssubscript𝑠1subscript𝑠s_{1},\dots,s_{\ell} are in the image of Bb×B𝔭×superscriptsubscript𝐵𝑏superscriptsubscript𝐵𝔭{B_{b}^{\times}}\to{B_{\mathfrak{p}}^{\times}}. We may replace B𝐵B, 𝔭𝔭{\mathfrak{p}} with Bbsubscript𝐵𝑏B_{b}, 𝔭bsubscript𝔭𝑏{\mathfrak{p}}_{b} and assume s1,,sB×subscript𝑠1subscript𝑠superscript𝐵s_{1},\dots,s_{\ell}\in{B^{\times}} henceforth.

Write B=B[s1,,s]superscript𝐵𝐵subscript𝑠1subscript𝑠B^{\prime}=B[\sqrt{s_{1}},\dots,\sqrt{s_{\ell}}] and A=A[s1,,s]superscript𝐴𝐴subscript𝑠1subscript𝑠A^{\prime}=A[\sqrt{s_{1}},\dots,\sqrt{s_{\ell}}], and choose g1,,grB𝔭subscript𝑔1subscript𝑔𝑟subscriptsuperscript𝐵𝔭g_{1},\dots,g_{r}\in B^{\prime}_{\mathfrak{p}} such that ifigi=1subscript𝑖subscript𝑓𝑖subscript𝑔𝑖1\sum_{i}f_{i}g_{i}=1. Then there is bB𝔭𝑏𝐵𝔭b\in B-{\mathfrak{p}} such that f1,,fr,g1,,grsubscript𝑓1subscript𝑓𝑟subscript𝑔1subscript𝑔𝑟f_{1},\dots,f_{r},g_{1},\dots,g_{r} are images of elements in Bbsubscriptsuperscript𝐵𝑏B^{\prime}_{b}, also denoted f1,,fr,g1,,grsubscript𝑓1subscript𝑓𝑟subscript𝑔1subscript𝑔𝑟f_{1},\dots,f_{r},g_{1},\dots,g_{r}, and such that igifi=1subscript𝑖subscript𝑔𝑖subscript𝑓𝑖1\sum_{i}g_{i}f_{i}=1 in Bbsubscriptsuperscript𝐵𝑏B^{\prime}_{b}. Again, we replace B𝐵B, 𝔭𝔭{\mathfrak{p}} with Bbsubscript𝐵𝑏B_{b}, 𝔭bsubscript𝔭𝑏{\mathfrak{p}}_{b} and assume f1,,frBsubscript𝑓1subscript𝑓𝑟superscript𝐵f_{1},\dots,f_{r}\in B^{\prime}.

Fix 1ir1𝑖𝑟1\leq i\leq r. There are viMn×n(A)subscriptsuperscript𝑣𝑖subscriptM𝑛𝑛superscript𝐴v^{\prime}_{i}\in\mathrm{M}_{n\times n}(A^{\prime}), bB𝔭𝑏𝐵𝔭b\in B-{\mathfrak{p}} and m{0}𝑚0m\in\mathbb{N}\cup\{0\} such that vi=vib1fimsubscript𝑣𝑖subscriptsuperscript𝑣𝑖superscript𝑏1superscriptsubscript𝑓𝑖𝑚v_{i}=v^{\prime}_{i}b^{-1}f_{i}^{-m} in Mn×n((A𝔭)fi)subscriptM𝑛𝑛subscriptsubscriptsuperscript𝐴𝔭subscript𝑓𝑖\mathrm{M}_{n\times n}((A^{\prime}_{\mathfrak{p}})_{f_{i}}). Since viλtrhvi=hsuperscriptsubscript𝑣𝑖𝜆trsubscript𝑣𝑖superscriptv_{i}^{\lambda\text{\rm tr}}hv_{i}=h^{\prime}, we have viλtrhvi=b2fi2mhsubscriptsuperscript𝑣𝜆tr𝑖subscriptsuperscript𝑣𝑖superscript𝑏2superscriptsubscript𝑓𝑖2𝑚superscriptv^{\prime\lambda\text{\rm tr}}_{i}hv^{\prime}_{i}=b^{2}f_{i}^{2m}h^{\prime} in Mn×n((A𝔭)fi)subscriptM𝑛𝑛subscriptsubscriptsuperscript𝐴𝔭subscript𝑓𝑖\mathrm{M}_{n\times n}((A^{\prime}_{\mathfrak{p}})_{f_{i}}), and hence there is k0𝑘0k\in\mathbb{N}\cup 0 such that fikviλtrhvi=b2fi2m+khsuperscriptsubscript𝑓𝑖𝑘subscriptsuperscript𝑣𝜆tr𝑖subscriptsuperscript𝑣𝑖superscript𝑏2superscriptsubscript𝑓𝑖2𝑚𝑘superscriptf_{i}^{k}v^{\prime\lambda\text{\rm tr}}_{i}hv^{\prime}_{i}=b^{2}f_{i}^{2m+k}h^{\prime} in Mn×n(A𝔭)subscriptM𝑛𝑛subscriptsuperscript𝐴𝔭\mathrm{M}_{n\times n}(A^{\prime}_{\mathfrak{p}}). This in turn implies that there is bB𝔭superscript𝑏𝐵𝔭b^{\prime}\in B-{\mathfrak{p}} such that bfikviλtrhvi=bb2fi2m+khsuperscript𝑏superscriptsubscript𝑓𝑖𝑘subscriptsuperscript𝑣𝜆tr𝑖subscriptsuperscript𝑣𝑖superscript𝑏superscript𝑏2superscriptsubscript𝑓𝑖2𝑚𝑘superscriptb^{\prime}f_{i}^{k}v^{\prime\lambda\text{\rm tr}}_{i}hv^{\prime}_{i}=b^{\prime}b^{2}f_{i}^{2m+k}h^{\prime} in Mn×n(A)subscriptM𝑛𝑛superscript𝐴\mathrm{M}_{n\times n}(A^{\prime}). Replacing B𝐵B, 𝔭𝔭{\mathfrak{p}} with Bbbsubscript𝐵𝑏superscript𝑏B_{bb^{\prime}}, 𝔭bbsubscript𝔭𝑏superscript𝑏{\mathfrak{p}}_{bb^{\prime}}, we may assume b,bB×𝑏superscript𝑏superscript𝐵b,b^{\prime}\in{B^{\times}}. Let v~i=vib1fimMn×n(Afi)subscript~𝑣𝑖subscriptsuperscript𝑣𝑖superscript𝑏1superscriptsubscript𝑓𝑖𝑚subscriptM𝑛𝑛subscriptsuperscript𝐴subscript𝑓𝑖\tilde{v}_{i}=v^{\prime}_{i}b^{-1}f_{i}^{-m}\in\mathrm{M}_{n\times n}(A^{\prime}_{f_{i}}). Then the image of v~isubscript~𝑣𝑖\tilde{v}_{i} in Mn×n((A𝔭)fi)subscriptM𝑛𝑛subscriptsubscriptsuperscript𝐴𝔭subscript𝑓𝑖\mathrm{M}_{n\times n}((A^{\prime}_{\mathfrak{p}})_{f_{i}}) is visubscript𝑣𝑖v_{i} and the equality bfikviλtrhvi=bb2fi2m+khsuperscript𝑏superscriptsubscript𝑓𝑖𝑘subscriptsuperscript𝑣𝜆tr𝑖subscriptsuperscript𝑣𝑖superscript𝑏superscript𝑏2superscriptsubscript𝑓𝑖2𝑚𝑘superscriptb^{\prime}f_{i}^{k}v^{\prime\lambda\text{\rm tr}}_{i}hv^{\prime}_{i}=b^{\prime}b^{2}f_{i}^{2m+k}h^{\prime} implies that v~iλtrhv~i=hsuperscriptsubscript~𝑣𝑖𝜆trsubscript~𝑣𝑖superscript\tilde{v}_{i}^{\lambda\text{\rm tr}}h\tilde{v}_{i}=h^{\prime} in Mn×n(Afi)subscriptM𝑛𝑛subscriptsuperscript𝐴subscript𝑓𝑖\mathrm{M}_{n\times n}(A^{\prime}_{f_{i}}). Taking determinants, we see that v~iGLn(Afi)subscript~𝑣𝑖subscriptGL𝑛subscriptsuperscript𝐴subscript𝑓𝑖\tilde{v}_{i}\in\operatorname{GL}_{n}(A^{\prime}_{f_{i}}).

The lemma follows by applying the previous paragraph to all 1ir1𝑖𝑟1\leq i\leq r. ∎

We now return to the context of ringed topoi.

Lemma 5.3.6.

Assume 2S×2superscript𝑆2\in{S^{\times}}, and let U𝐘𝑈𝐘U\in{\mathbf{Y}}, εN(U)𝜀𝑁𝑈\varepsilon\in N(U) and t=ε¯T(U)𝑡¯𝜀𝑇𝑈t=\overline{\varepsilon}\in T(U). Then:

  1. (i)

    There is a covering {ViU}i=1,2subscriptsubscript𝑉𝑖𝑈𝑖12\{{V_{i}\to U}\}_{i=1,2} of U𝑈U such that 1+εR×(V1)1𝜀superscript𝑅subscript𝑉11+\varepsilon\in{R^{\times}}(V_{1}) and 1εR×(V2)1𝜀superscript𝑅subscript𝑉21-\varepsilon\in{R^{\times}}(V_{2}).

  2. (ii)

    There exists a covering {ViU}i=1,2subscriptsubscript𝑉𝑖𝑈𝑖12\{V_{i}\to U\}_{i=1,2} of U𝑈U such that t|V1=1evaluated-at𝑡subscript𝑉11t|_{V_{1}}=1 and t|V2=1evaluated-at𝑡subscript𝑉21t|_{V_{2}}=-1. Equivalently, there exists a covering {ViU}i=1,2subscriptsubscript𝑉𝑖𝑈𝑖12\{V_{i}\to U\}_{i=1,2} of U𝑈U and βiR(Vi)subscript𝛽𝑖𝑅subscript𝑉𝑖\beta_{i}\in R(V_{i}) (i=1,2𝑖12i=1,2) such that β11β1λ=ε|V1superscriptsubscript𝛽11superscriptsubscript𝛽1𝜆evaluated-at𝜀subscript𝑉1\beta_{1}^{-1}\beta_{1}^{\lambda}=\varepsilon|_{V_{1}} and β21β2λ=ε|V2superscriptsubscript𝛽21superscriptsubscript𝛽2𝜆evaluated-at𝜀subscript𝑉2-\beta_{2}^{-1}\beta_{2}^{\lambda}=\varepsilon|_{V_{2}}.

Proof.
  1. (i)

    We note that this statement requires proof because R𝑅R is not a local ring object in general. Observe that ε1(1±ε)2=ελ±2+εsuperscript𝜀1superscriptplus-or-minus1𝜀2plus-or-minussuperscript𝜀𝜆2𝜀\varepsilon^{-1}(1\pm\varepsilon)^{2}=\varepsilon^{\lambda}\pm 2+\varepsilon and hence ε1(1±ε)2S(U)superscript𝜀1superscriptplus-or-minus1𝜀2𝑆𝑈\varepsilon^{-1}(1\pm\varepsilon)^{2}\in S(U). Since ε1(1+ε)2ε1(1ε)2=4superscript𝜀1superscript1𝜀2superscript𝜀1superscript1𝜀24\varepsilon^{-1}(1+\varepsilon)^{2}-\varepsilon^{-1}(1-\varepsilon)^{2}=4 and S𝑆S is a local ring object, the assumption 2S×2superscript𝑆2\in{S^{\times}} implies that there exists a covering {ViU}i=1,2subscriptsubscript𝑉𝑖𝑈𝑖12\{V_{i}\to U\}_{i=1,2} of U𝑈U such that ε1(1+ε)2S×(V1)superscript𝜀1superscript1𝜀2superscript𝑆subscript𝑉1\varepsilon^{-1}(1+\varepsilon)^{2}\in{S^{\times}}(V_{1}) and ε1(1ε)2S×(V2)superscript𝜀1superscript1𝜀2superscript𝑆subscript𝑉2\varepsilon^{-1}({1-\varepsilon})^{2}\in{S^{\times}}(V_{2}). It follows that 1+εR×(V1)1𝜀superscript𝑅subscript𝑉11+\varepsilon\in{R^{\times}}(V_{1}) and 1εR×(V2)1𝜀superscript𝑅subscript𝑉21-\varepsilon\in{R^{\times}}(V_{2}).

  2. (ii)

    Choose a covering {ViU}i=1,2subscriptsubscript𝑉𝑖𝑈𝑖12\{V_{i}\to U\}_{i=1,2} as in (i) and let β1=(1+ε)|V1subscript𝛽1evaluated-at1𝜀subscript𝑉1\beta_{1}=(1+\varepsilon)|_{V_{1}}, β2=(1ε)|V2subscript𝛽2evaluated-at1𝜀subscript𝑉2\beta_{2}=(1-\varepsilon)|_{V_{2}}. Since ελ=ε1superscript𝜀𝜆superscript𝜀1\varepsilon^{\lambda}=\varepsilon^{-1}, we have β1λε=β1superscriptsubscript𝛽1𝜆𝜀subscript𝛽1\beta_{1}^{\lambda}\varepsilon=\beta_{1}, and hence β11β1λ=εsuperscriptsubscript𝛽11superscriptsubscript𝛽1𝜆𝜀\beta_{1}^{-1}\beta_{1}^{\lambda}=\varepsilon. Likewise, β21β2λ=εsuperscriptsubscript𝛽21superscriptsubscript𝛽2𝜆𝜀\beta_{2}^{-1}\beta_{2}^{\lambda}=-\varepsilon. ∎

The following lemma is known when π:𝐗𝐘:𝜋𝐗𝐘\pi:{\mathbf{X}}\to{\mathbf{Y}} is unramified or trivial, i.e., when R𝑅R is a quadratic étale S𝑆S-algebra or R=S𝑅𝑆R=S. The ramified situation that we consider appears not to have been considered before in the literature.

Lemma 5.3.7.

Let U𝐘𝑈𝐘U\in{\mathbf{Y}}, let εN(U)𝜀𝑁𝑈\varepsilon\in N(U), and let h,hGLn(R)(U)superscriptsubscriptGL𝑛𝑅𝑈h,h^{\prime}\in\operatorname{GL}_{n}(R)(U) be two (ε,λtr)𝜀𝜆tr(\varepsilon,\lambda\text{\rm tr})-hermitian matrices, i.e., h=εhλtr𝜀superscript𝜆trh=\varepsilon h^{\lambda\text{\rm tr}} and h=εhλtrsuperscript𝜀superscript𝜆trh^{\prime}=\varepsilon h^{\prime\lambda\text{\rm tr}}. Assume S×superscript𝑆{S^{\times}} has square roots locally and that 2S×2superscript𝑆2\in{S^{\times}}, or π𝜋\pi is unramified, or n𝑛n is odd. Then there exists a covering VU𝑉𝑈V\to U and vGLn(R)(V)𝑣subscriptGL𝑛𝑅𝑉v\in\operatorname{GL}_{n}(R)(V) such that vλtrhv=hsuperscript𝑣𝜆tr𝑣superscriptv^{\lambda\text{\rm tr}}hv=h^{\prime} in GLn(R)(V)subscriptGL𝑛𝑅𝑉\operatorname{GL}_{n}(R)(V).

Proof.

Suppose first that 2S×2superscript𝑆2\in{S^{\times}}. Let {ViU}i=1,2subscriptsubscript𝑉𝑖𝑈𝑖12\{V_{i}\to U\}_{i=1,2} and β1,β2subscript𝛽1subscript𝛽2\beta_{1},\beta_{2} be as in Lemma 5.3.6(ii). We may replace h,h,Usuperscript𝑈h,h^{\prime},U with (βih,βih,Vi)i=1,2subscriptsubscript𝛽𝑖subscript𝛽𝑖superscriptsubscript𝑉𝑖𝑖12(\beta_{i}h,\beta_{i}h^{\prime},V_{i})_{i=1,2} and assume that ε{±1}𝜀plus-or-minus1\varepsilon\in\{\pm 1\} henceforth.

We claim that it is enough to show that for all 𝔭SpecS(U)𝔭Spec𝑆𝑈{\mathfrak{p}}\in\operatorname{Spec}S(U), there are s1,,sS(U)𝔭×subscript𝑠1subscript𝑠𝑆superscriptsubscript𝑈𝔭s_{1},\dots,s_{\ell}\in{S(U)_{\mathfrak{p}}^{\times}}, f1,,frS(U)𝔭[s1,,s]subscript𝑓1subscript𝑓𝑟𝑆subscript𝑈𝔭subscript𝑠1subscript𝑠f_{1},\dots,f_{r}\in S(U)_{\mathfrak{p}}[\sqrt{s_{1}},\dots,\sqrt{s_{\ell}}] and vjGLn(R(U)𝔭[s1,,s]fj)subscript𝑣𝑗subscriptGL𝑛𝑅subscript𝑈𝔭subscriptsubscript𝑠1subscript𝑠subscript𝑓𝑗v_{j}\in\operatorname{GL}_{n}(R(U)_{\mathfrak{p}}[\sqrt{s_{1}},\dots,\sqrt{s_{\ell}}]_{f_{j}}) (j=1,,r𝑗1𝑟j=1,\dots,r) such that f1,,frsubscript𝑓1subscript𝑓𝑟f_{1},\dots,f_{r} generate the unit ideal in S(U)𝔭[s1,,s]𝑆subscript𝑈𝔭subscript𝑠1subscript𝑠S(U)_{\mathfrak{p}}[\sqrt{s_{1}},\dots,\sqrt{s_{\ell}}] and vjλtrhvj=hsuperscriptsubscript𝑣𝑗𝜆trsubscript𝑣𝑗superscriptv_{j}^{\lambda\text{\rm tr}}hv_{j}=h^{\prime}. If this holds, then Lemma 5.3.5 implies that for all 𝔭𝔭{\mathfrak{p}}, we can find b𝔭S(U)𝔭subscript𝑏𝔭𝑆𝑈𝔭b_{\mathfrak{p}}\in S(U)-{\mathfrak{p}} such that the previous condition holds upon replacing S(U)𝔭,R(U)𝔭𝑆subscript𝑈𝔭𝑅subscript𝑈𝔭S(U)_{\mathfrak{p}},R(U)_{\mathfrak{p}} with S(U)b𝔭,R(U)b𝔭𝑆subscript𝑈subscript𝑏𝔭𝑅subscript𝑈subscript𝑏𝔭S(U)_{b_{\mathfrak{p}}},R(U)_{b_{\mathfrak{p}}}. Since 𝔭b𝔭S(U)=S(U)subscript𝔭subscript𝑏𝔭𝑆𝑈𝑆𝑈\sum_{\mathfrak{p}}b_{\mathfrak{p}}S(U)=S(U) and S𝑆S is a local ring object, there is a covering {V𝔭U}𝔭subscriptsubscript𝑉𝔭𝑈𝔭\{V_{\mathfrak{p}}\to U\}_{\mathfrak{p}} such that b𝔭S(V𝔭)×subscript𝑏𝔭𝑆superscriptsubscript𝑉𝔭b_{\mathfrak{p}}\in{S(V_{\mathfrak{p}})^{\times}}. Fix some 𝔭𝔭{\mathfrak{p}} and let b𝔭,s1,,s,f1,,fr,v1,,vrsubscript𝑏𝔭subscript𝑠1subscript𝑠subscript𝑓1subscript𝑓𝑟subscript𝑣1subscript𝑣𝑟b_{\mathfrak{p}},s_{1},\dots,s_{\ell},f_{1},\dots,f_{r},v_{1},\dots,v_{r} be as above. By construction, the images of s1,,ssubscript𝑠1subscript𝑠s_{1},\dots,s_{\ell} in S(V𝔭)𝑆subscript𝑉𝔭S(V_{\mathfrak{p}}) are invertible in S(V𝔭)𝑆subscript𝑉𝔭S(V_{\mathfrak{p}}). Since S×superscript𝑆{S^{\times}} has square roots locally, we can replace V𝔭subscript𝑉𝔭V_{\mathfrak{p}} with a suitable covering such that s1,,ssubscript𝑠1subscript𝑠s_{1},\dots,s_{\ell} have square roots in S(V𝔭)𝑆subscript𝑉𝔭S(V_{\mathfrak{p}}). In particular, S(U)S(V𝔭)𝑆𝑈𝑆subscript𝑉𝔭S(U)\to S(V_{\mathfrak{p}}) factors through S(U)S(U)b𝔭[s1,,s]𝑆𝑈𝑆subscript𝑈subscript𝑏𝔭subscript𝑠1subscript𝑠S(U)\to S(U)_{b_{\mathfrak{p}}}[\sqrt{s_{1}},\dots,\sqrt{s_{\ell}}]. Applying the fact that S𝑆S is a local ring object again, we see that there is a covering {V𝔭,iV𝔭}i=1rsuperscriptsubscriptsubscript𝑉𝔭𝑖subscript𝑉𝔭𝑖1𝑟\{V_{{\mathfrak{p}},i}\to V_{\mathfrak{p}}\}_{i=1}^{r} such that the image of fisubscript𝑓𝑖f_{i} in S(V𝔭,i)𝑆subscript𝑉𝔭𝑖{S(V_{{\mathfrak{p}},i})} is invertible. It follows that R(U)R(V𝔭,i)𝑅𝑈𝑅subscript𝑉𝔭𝑖R(U)\to R(V_{{\mathfrak{p}},i}) factors through R(U)b𝔭[s1,,ss]fi𝑅subscript𝑈subscript𝑏𝔭subscriptsubscript𝑠1subscript𝑠subscript𝑠subscript𝑓𝑖R(U)_{b_{\mathfrak{p}}}[\sqrt{s_{1}},\dots,s_{s_{\ell}}]_{f_{i}} and hence there is v𝔭,iGLn(R(V𝔭,i))subscript𝑣𝔭𝑖subscriptGL𝑛𝑅subscript𝑉𝔭𝑖v_{{\mathfrak{p}},i}\in\operatorname{GL}_{n}(R(V_{{\mathfrak{p}},i})) — the image of visubscript𝑣𝑖v_{i} — such that v𝔭,iλtrhv𝔭,i=hsuperscriptsubscript𝑣𝔭𝑖𝜆trsubscript𝑣𝔭𝑖superscriptv_{{\mathfrak{p}},i}^{\lambda\text{\rm tr}}hv_{{\mathfrak{p}},i}=h^{\prime}. Finally, let V=𝔭,iV𝔭,i𝑉subscriptsquare-union𝔭𝑖subscript𝑉𝔭𝑖V=\bigsqcup_{{\mathfrak{p}},i}V_{{\mathfrak{p}},i} and take v=(v𝔭,i)𝔭,iGLn(R(V))=𝔭,iGLn(R(V𝔭,i))𝑣subscriptsubscript𝑣𝔭𝑖𝔭𝑖subscriptGL𝑛𝑅𝑉subscriptproduct𝔭𝑖subscriptGL𝑛𝑅subscript𝑉𝔭𝑖v=(v_{{\mathfrak{p}},i})_{{\mathfrak{p}},i}\in\operatorname{GL}_{n}(R(V))=\prod_{{\mathfrak{p}},i}\operatorname{GL}_{n}(R(V_{{\mathfrak{p}},i})).

Let 𝔭SpecS(U)𝔭Spec𝑆𝑈{\mathfrak{p}}\in\operatorname{Spec}S(U). We now prove the existence of s1,,s,f1,,fr,v1,,vrsubscript𝑠1subscript𝑠subscript𝑓1subscript𝑓𝑟subscript𝑣1subscript𝑣𝑟s_{1},\dots,s_{\ell},f_{1},\dots,f_{r},v_{1},\dots,v_{r} above. Write B=S(U)𝔭𝐵𝑆subscript𝑈𝔭B=S(U)_{\mathfrak{p}} and A=R(U)𝔭𝐴𝑅subscript𝑈𝔭A=R(U)_{\mathfrak{p}}. Then B𝐵B is local and it is the fixed ring of λ:AA:𝜆𝐴𝐴\lambda:A\to A. We write A¯=A/Jac(A)¯𝐴𝐴Jac𝐴\overline{A}=A/\mathrm{Jac}(A) and let λ¯:A¯A¯:¯𝜆¯𝐴¯𝐴\overline{\lambda}:\overline{A}\to\overline{A} denote the involution induced by λ𝜆\lambda.

Suppose ε=1𝜀1\varepsilon=1 or λ¯id¯𝜆id\overline{\lambda}\neq\mathrm{id}. By Lemma 5.3.1, we may assume that hh and hsuperscripth^{\prime} are diagonal, say h=diag(α1,,αn)diagsubscript𝛼1subscript𝛼𝑛h=\mathrm{diag}(\alpha_{1},\dots,\alpha_{n}) and hdiag(α1,,αn)superscriptdiagsubscriptsuperscript𝛼1subscriptsuperscript𝛼𝑛h^{\prime}\in\mathrm{diag}(\alpha^{\prime}_{1},\dots,\alpha^{\prime}_{n}). Since hh and hsuperscripth^{\prime} are (ε,λtr)𝜀𝜆tr(\varepsilon,\lambda\text{\rm tr})-hermitian, we have (αi1αi)λ=αi1εαiε1=αi1αisuperscriptsuperscriptsubscript𝛼𝑖1subscriptsuperscript𝛼𝑖𝜆superscriptsubscript𝛼𝑖1𝜀subscriptsuperscript𝛼𝑖superscript𝜀1superscriptsubscript𝛼𝑖1superscriptsubscript𝛼𝑖(\alpha_{i}^{-1}\alpha^{\prime}_{i})^{\lambda}=\alpha_{i}^{-1}\varepsilon\alpha^{\prime}_{i}\varepsilon^{-1}=\alpha_{i}^{-1}\alpha_{i}^{\prime}, hence αi1αiB×superscriptsubscript𝛼𝑖1subscriptsuperscript𝛼𝑖superscript𝐵\alpha_{i}^{-1}\alpha^{\prime}_{i}\in{B^{\times}} for all i𝑖i. Writing si=αi1α1subscript𝑠𝑖superscriptsubscript𝛼𝑖1subscriptsuperscript𝛼1s_{i}=\alpha_{i}^{-1}\alpha^{\prime}_{1} and v=diag(s1,,sn)GLn(A[s1,,sn])𝑣diagsubscript𝑠1subscript𝑠𝑛subscriptGL𝑛𝐴subscript𝑠1subscript𝑠𝑛v=\mathrm{diag}(\sqrt{s_{1}},\dots,\sqrt{s_{n}})\in\operatorname{GL}_{n}(A[\sqrt{s_{1}},\dots,\sqrt{s_{n}}]), we have vλtrhv=hsuperscript𝑣𝜆tr𝑣superscriptv^{\lambda\text{\rm tr}}hv=h^{\prime}, as required (take f1=1subscript𝑓11f_{1}=1).

Suppose now that ε=1𝜀1\varepsilon=-1 and λ¯=id¯𝜆id\overline{\lambda}=\mathrm{id}. Applying Lemma 5.3.2 to hh and hsuperscripth^{\prime}, we may assume that hh and hsuperscripth^{\prime} are direct sums of 2×2222\times 2 matrices in [0110]+M2×2(Jac(A))delimited-[]0110subscriptM22Jac𝐴[\begin{smallmatrix}0&1\\ -1&0\end{smallmatrix}]+\mathrm{M}_{2\times 2}(\mathrm{Jac}(A)), say h=h1hmdirect-sumsubscript1subscript𝑚h=h_{1}\oplus\dots\oplus h_{m}, h=hm+1hnsuperscriptdirect-sumsubscript𝑚1subscript𝑛h^{\prime}=h_{m+1}\oplus\dots\oplus h_{n} with m=n/2𝑚𝑛2m=n/2. Applying Lemma 5.3.4 to hjsubscript𝑗h_{j}, we obtain sj,tjB×subscript𝑠𝑗subscript𝑡𝑗superscript𝐵s_{j},t_{j}\in{B^{\times}}, fj1,,fjrjB[sj,tj]subscript𝑓𝑗1subscript𝑓𝑗subscript𝑟𝑗𝐵subscript𝑠𝑗subscript𝑡𝑗f_{j1},\dots,f_{jr_{j}}\in B[\sqrt{s_{j}},\sqrt{t_{j}}], and vjiGL2(A[sj,tj]fji)subscript𝑣𝑗𝑖subscriptGL2𝐴subscriptsubscript𝑠𝑗subscript𝑡𝑗subscript𝑓𝑗𝑖v_{ji}\in\operatorname{GL}_{2}(A[\sqrt{s_{j}},\sqrt{t_{j}}]_{f_{ji}}) such that vjiλtrhjvji=[0110]superscriptsubscript𝑣𝑗𝑖𝜆trsubscript𝑗subscript𝑣𝑗𝑖delimited-[]0110v_{ji}^{\lambda\text{\rm tr}}h_{j}v_{ji}=[\begin{smallmatrix}0&1\\ -1&0\end{smallmatrix}].

Let B=B[s1,t1,,sn,tn]superscript𝐵𝐵subscript𝑠1subscript𝑡1subscript𝑠𝑛subscript𝑡𝑛B^{\prime}=B[\sqrt{s_{1}},\sqrt{t_{1}},\dots,\sqrt{s_{n}},\sqrt{t_{n}}], A=A[s1,t1,,sn,tn]superscript𝐴𝐴subscript𝑠1subscript𝑡1subscript𝑠𝑛subscript𝑡𝑛A^{\prime}=A[\sqrt{s_{1}},\sqrt{t_{1}},\dots,\sqrt{s_{n}},\sqrt{t_{n}}] and regard {fji}j,isubscriptsubscript𝑓𝑗𝑖𝑗𝑖\{f_{ji}\}_{j,i} as elements of Bsuperscript𝐵B^{\prime}. For every tuple I=(i1,,in)j{1,,rj}𝐼subscript𝑖1subscript𝑖𝑛subscriptproduct𝑗1subscript𝑟𝑗I=(i_{1},\dots,i_{n})\in\prod_{j}\{1,\dots,r_{j}\}, let fI=jfjijsubscript𝑓𝐼subscriptproduct𝑗subscript𝑓𝑗subscript𝑖𝑗f_{I}=\prod_{j}f_{ji_{j}}, vI=v1i1vmimsubscript𝑣𝐼direct-sumsubscript𝑣1subscript𝑖1subscript𝑣𝑚subscript𝑖𝑚v_{I}=v_{1i_{1}}\oplus\dots\oplus v_{mi_{m}} and vI=v(m+1)im+1vninsubscriptsuperscript𝑣𝐼direct-sumsubscript𝑣𝑚1subscript𝑖𝑚1subscript𝑣𝑛subscript𝑖𝑛v^{\prime}_{I}=v_{(m+1)i_{m+1}}\oplus\dots\oplus v_{ni_{n}}, where vI,vIsubscript𝑣𝐼subscriptsuperscript𝑣𝐼v_{I},v^{\prime}_{I} are regarded as elements of GLn(AfI)subscriptGL𝑛subscriptsuperscript𝐴subscript𝑓𝐼\operatorname{GL}_{n}(A^{\prime}_{f_{I}}). Then IfIB=j(i=1rjfjiB)=Bsubscript𝐼subscript𝑓𝐼superscript𝐵subscriptproduct𝑗superscriptsubscript𝑖1subscript𝑟𝑗subscript𝑓𝑗𝑖superscript𝐵superscript𝐵\sum_{I}f_{I}B^{\prime}=\prod_{j}(\sum_{i=1}^{r_{j}}f_{ji}B^{\prime})=B^{\prime} and (vIvI1)λtrh(vIvI1)=hsuperscriptsubscript𝑣𝐼subscriptsuperscript𝑣1𝐼𝜆trsubscript𝑣𝐼subscriptsuperscript𝑣1𝐼superscript(v_{I}v^{\prime-1}_{I})^{\lambda\text{\rm tr}}h(v_{I}v^{\prime-1}_{I})=h^{\prime}, which is what we want. This establishes the lemma when 2S×2superscript𝑆2\in{S^{\times}}.

To prove the remaining cases, we observe that when π𝜋\pi is unramified, or n𝑛n is odd, the use of Lemmas 5.3.2, 5.3.4 and 5.3.6 can be avoided, and hence the assumption 2S×2superscript𝑆2\in{S^{\times}} is unnecessary.

When π𝜋\pi is unramified, we apply Proposition 5.2.2 instead of Lemma 5.3.6(ii) and assume ε=1𝜀1\varepsilon=1 hereafter. Since in this case A𝐴A is a quadratic étale B𝐵B-algebra, Proposition 3.1.4(ii) implies that λ¯id¯𝜆id\overline{\lambda}\neq\mathrm{id}, and so the case λ¯=id¯𝜆id\overline{\lambda}=\mathrm{id} does not occur.

Suppose now that n𝑛n is odd, say n=2m+1𝑛2𝑚1n=2m+1. Taking the determinant of both sides of h=εhλtr𝜀superscript𝜆trh=\varepsilon h^{\lambda\text{\rm tr}} yields deth=ε2m+1(deth)λsuperscript𝜀2𝑚1superscript𝜆\det h=\varepsilon^{2m+1}(\det h)^{\lambda}. Since ελ=ε1superscript𝜀𝜆superscript𝜀1\varepsilon^{\lambda}=\varepsilon^{-1}, this implies that ε=β1βλ𝜀superscript𝛽1superscript𝛽𝜆\varepsilon=\beta^{-1}\beta^{\lambda} for β=εm(deth)λ𝛽superscript𝜀𝑚superscript𝜆\beta=\varepsilon^{m}(\det h)^{\lambda}. Replacing h,hsuperscripth,h^{\prime} with βh,βh𝛽𝛽superscript\beta h,\beta h^{\prime}, we may assume ε=1𝜀1\varepsilon=1. Since n𝑛n is odd, we can now apply Lemma 5.3.1 even when λ¯=id¯𝜆id\overline{\lambda}=\mathrm{id} and finish the proof without using Lemmas 5.3.2, 5.3.4 or 5.3.6. ∎

We can finally complete the proof of Theorem 5.2.13.

Proposition 5.3.8.

Let n𝑛n be a positive integer. Suppose S×superscript𝑆{S^{\times}} has square roots locally and at least one of the following holds:

  1. (1)

    2S×2superscript𝑆2\in S^{\times}.

  2. (2)

    π:𝐗𝐘:𝜋𝐗𝐘\pi:\mathbf{X}\to\mathbf{Y} is unramified.

  3. (3)

    n𝑛n is odd.

Let (A,τ)𝐴𝜏(A,\tau) and (A,τ)superscript𝐴superscript𝜏(A^{\prime},\tau^{\prime}) be two degree-n𝑛n Azumaya R𝑅R-algebras with λ𝜆\lambda-involutions having the same coarse type. Then (A,τ)𝐴𝜏(A,\tau) and (A,τ)superscript𝐴superscript𝜏(A^{\prime},\tau^{\prime}) are locally isomorphic as R𝑅R-algebras with involution.

Proof.

Following the construction of ct(A,τ)ct𝐴𝜏\mathrm{ct}(A,\tau) in 5.2, define U𝑈U, ψ𝜓\psi, σ𝜎\sigma, g𝑔g, hh and ε=hλtrhN(U)𝜀superscript𝜆tr𝑁𝑈\varepsilon=h^{-\lambda\text{\rm tr}}h\in N(U) so that ε¯T(U)¯𝜀𝑇𝑈\overline{\varepsilon}\in T(U) induces ct(A,τ)H0(T)ct𝐴𝜏superscriptH0𝑇\mathrm{ct}(A,\tau)\in\mathrm{H}^{0}(T). Repeating the construction with (A,τ)superscript𝐴superscript𝜏(A^{\prime},\tau^{\prime}) in place of (A,τ)𝐴𝜏(A,\tau), we define U,ψ,σ,g,h,εsuperscript𝑈superscript𝜓superscript𝜎superscript𝑔superscriptsuperscript𝜀U^{\prime},\psi^{\prime},\sigma^{\prime},g^{\prime},h^{\prime},\varepsilon^{\prime} analogously. By refining both U𝑈U and Usuperscript𝑈U^{\prime}, we may assume U=U𝑈superscript𝑈U=U^{\prime}.

Since ct(A,τ)=ct(A,τ)ct𝐴𝜏ctsuperscript𝐴superscript𝜏\mathrm{ct}(A,\tau)=\mathrm{ct}(A^{\prime},\tau^{\prime}), ε𝜀\varepsilon and εsuperscript𝜀\varepsilon^{\prime} determine the same section in T(U)𝑇𝑈T(U). Thus, there exists a covering VU𝑉𝑈V\to U and βR(V)𝛽𝑅𝑉\beta\in R(V) such that ε=β1βλεsuperscript𝜀superscript𝛽1superscript𝛽𝜆𝜀\varepsilon^{\prime}=\beta^{-1}\beta^{\lambda}\varepsilon. Replacing U𝑈U with V𝑉V, and hsuperscripth^{\prime} with βh𝛽superscript\beta h^{\prime}, we may assume that ε=εsuperscript𝜀𝜀\varepsilon^{\prime}=\varepsilon.

Now, by Lemma 5.3.7, there exists a covering VU𝑉𝑈V\to U and vGLn(R(V))𝑣subscriptGL𝑛𝑅𝑉v\in\operatorname{GL}_{n}(R(V)) such that vλtrhv=hsuperscript𝑣𝜆tr𝑣superscriptv^{\lambda\text{\rm tr}}hv=h^{\prime}. Again, replace U𝑈U with V𝑉V. Letting u𝑢u denote the image of v𝑣v in PGLn(R)(U)subscriptPGL𝑛𝑅𝑈\operatorname{PGL}_{n}(R)(U), we deduce gu=uλtrg𝑔𝑢superscript𝑢𝜆trsuperscript𝑔gu=u^{-\lambda\text{\rm tr}}g^{\prime}. Unfolding the construction in 5.2, one finds that τU=ψ1σψ=ψ1λtrgψsubscript𝜏𝑈superscript𝜓1𝜎𝜓superscript𝜓1𝜆tr𝑔𝜓\tau_{U}=\psi^{-1}\circ\sigma\circ\psi=\psi^{-1}\circ\lambda\text{\rm tr}\circ g\circ\psi, and likewise τU=ψ1λtrgψsubscriptsuperscript𝜏𝑈superscript𝜓1𝜆trsuperscript𝑔superscript𝜓\tau^{\prime}_{U}=\psi^{\prime-1}\circ\lambda\text{\rm tr}\circ g^{\prime}\circ\psi^{\prime}. Let θ:=ψ1uψ:AUAU:assign𝜃superscript𝜓1𝑢superscript𝜓subscriptsuperscript𝐴𝑈subscript𝐴𝑈\theta:=\psi^{-1}\circ u\circ\psi^{\prime}:A^{\prime}_{U}\to A_{U}. Then θ𝜃\theta is an isomorphism of R𝑅R-algebras, and since gu=uλtrg𝑔𝑢superscript𝑢𝜆trsuperscript𝑔gu=u^{-\lambda\text{\rm tr}}g^{\prime}, we have

θτU𝜃subscriptsuperscript𝜏𝑈\displaystyle\theta\circ\tau^{\prime}_{U} =ψ1uψψ1λtrgψabsentsuperscript𝜓1𝑢superscript𝜓superscript𝜓1𝜆trsuperscript𝑔superscript𝜓\displaystyle=\psi^{-1}\circ u\circ\psi^{\prime}\circ\psi^{\prime-1}\circ\lambda\text{\rm tr}\circ g^{\prime}\circ\psi^{\prime}
=ψ1λtruλtrgψabsentsuperscript𝜓1𝜆trsuperscript𝑢𝜆trsuperscript𝑔superscript𝜓\displaystyle=\psi^{-1}\circ\lambda\text{\rm tr}\circ u^{-\lambda\text{\rm tr}}g^{\prime}\circ\psi^{\prime}
=ψ1λtrguψabsentsuperscript𝜓1𝜆tr𝑔𝑢superscript𝜓\displaystyle=\psi^{-1}\circ\lambda\text{\rm tr}\circ gu\circ\psi^{\prime}
=ψ1λtrgψψ1uψ=τUθ.absentsuperscript𝜓1𝜆tr𝑔𝜓superscript𝜓1𝑢superscript𝜓subscript𝜏𝑈𝜃\displaystyle=\psi^{-1}\circ\lambda\text{\rm tr}\circ g\circ\psi\circ\psi^{-1}\circ u\circ\psi^{\prime}=\tau_{U}\circ\theta\ .

Thus, θ𝜃\theta defines an isomorphism of algebras with involution (AU,τU)(AU,τU)similar-tosubscriptsuperscript𝐴𝑈subscriptsuperscript𝜏𝑈subscript𝐴𝑈subscript𝜏𝑈(A^{\prime}_{U},\tau^{\prime}_{U})\xrightarrow{\sim}(A_{U},\tau_{U}). ∎

5.4. Determining types in specific cases

Under mild assumptions, Theorem 5.2.13 provides a cohomological criterion to determine whether two λ𝜆\lambda-involutions of Azumaya algebras have the same type, and Corollary 5.2.14 embeds the possible λ𝜆\lambda-types in cTyp(λ)=H0(T)cTyp𝜆superscriptH0𝑇{\mathrm{cTyp}({\lambda})}=\mathrm{H}^{0}(T). We finish this section by making this criterion and the realization of the types even more explicit in case the exact quotient π:𝐗𝐘:𝜋𝐗𝐘\pi:{\mathbf{X}}\to{\mathbf{Y}} is induced by a C2subscript𝐶2C_{2}-quotient of schemes or topological spaces.

Notation 5.4.1.

Throughout, we assume one of the following:

  1. (1)

    X𝑋X is a scheme on which 222 is invertible, λ:XX:𝜆𝑋𝑋\lambda:X\to X is an involution and π:XY:𝜋𝑋𝑌\pi:X\to Y is a good quotient relative to C2={1,λ}subscript𝐶21𝜆C_{2}=\{1,\lambda\}, see Example 4.3.3.

  2. (2)

    X𝑋X is a Hausdorff topological space, λ:XX:𝜆𝑋𝑋\lambda:X\to X is a continuous involution, and π:XY=X/{1,λ}:𝜋𝑋𝑌𝑋1𝜆\pi:X\to Y=X/\{1,\lambda\} is the quotient map.

We will usually treat both cases simultaneously, but when there is need to distinguish them, we shall address them as the scheme-theoretic case and the topological case, respectively.

In the scheme-theoretic case, the terms sheaf, cohomology and covering should be understood as étale sheaf, étale cohomology and étale covering, whereas in the topological case, they retain their ordinary meaning relative to the relevant topological space. Furthermore, in the topological case, 𝒪Xsubscript𝒪𝑋{\mathcal{O}}_{X} stands for 𝒞(X,)𝒞𝑋{\mathcal{C}}(X,\mathbb{C}), the sheaf of continuous functions into \mathbb{C}, and likewise for all topological spaces.

As in Subsection 5.2, write S=𝒪Y𝑆subscript𝒪𝑌S={\mathcal{O}}_{Y} and R=π𝒪X𝑅subscript𝜋subscript𝒪𝑋R=\pi_{*}{\mathcal{O}}_{X}, and define N𝑁N to be the kernel of the λ𝜆\lambda-norm xxλx:R×S×:maps-to𝑥superscript𝑥𝜆𝑥superscript𝑅superscript𝑆x\mapsto x^{\lambda}x:{R^{\times}}\to{S^{\times}} and T𝑇T to be the cokernel of xx1xλ:R×N:maps-to𝑥superscript𝑥1superscript𝑥𝜆superscript𝑅𝑁x\mapsto x^{-1}x^{\lambda}:{R^{\times}}\to N. By means of Theorem 4.4.4, the results of the previous subsections can be applied, essentially verbatim, to Azumaya 𝒪Xsubscript𝒪𝑋{\mathcal{O}}_{X}-algebras with λ𝜆\lambda-involution.

Recall from Propositions 4.5.3 and 4.5.4 that there is a maximal open subscheme, resp. subset, UY𝑈𝑌U\subseteq Y such that πU:π1(U)U:subscript𝜋𝑈superscript𝜋1𝑈𝑈\pi_{U}:\pi^{-1}(U)\to U is unramified, i.e. a quadratic étale morphism or a double covering of topological spaces. We write

W=YUandZ=π1(W).formulae-sequence𝑊𝑌𝑈and𝑍superscript𝜋1𝑊W=Y-U\qquad\text{and}\qquad Z=\pi^{-1}(W)\ .

Then W𝑊W and Z𝑍Z are the branch locus and the ramification locus of π:XY:𝜋𝑋𝑌\pi:X\to Y, respectively. We endow Z𝑍Z and W𝑊W with the subspace topologies. In the scheme-theoretic case, we further endow them with the reduced induced closed subscheme structures in X𝑋X and Y𝑌Y, respectively. Recall from Proposition 4.5.5 and the preceding comment that π𝜋\pi induces an isomorphism of schemes, resp. topological spaces, ZW𝑍𝑊Z\to W, and λ𝜆\lambda restricts to the identity map on Z𝑍Z. In particular, W=Z/C2𝑊𝑍subscript𝐶2W=Z/C_{2}.

Recall that μ2,𝒪Wsubscript𝜇2subscript𝒪𝑊\mu_{2,{\mathcal{O}}_{W}} denotes the sheaf of square roots of 111 in 𝒪Wsubscript𝒪𝑊{\mathcal{O}}_{W}; we abbreviate this sheaf as μ2,Wsubscript𝜇2𝑊\mu_{2,W}. Since 222 is invertible in 𝒪Wsubscript𝒪𝑊{\mathcal{O}}_{W}, the sheaf μ2,Wsubscript𝜇2𝑊\mu_{2,W} is just the constant sheaf {±1}plus-or-minus1\{\pm 1\}. Similar notation applies to Z𝑍Z.

Let (A,τ)𝐴𝜏(A,\tau) be an Azumaya 𝒪Xsubscript𝒪𝑋{\mathcal{O}}_{X}-algebra with λ𝜆\lambda-involution and let zZ𝑧𝑍z\in Z be a point of the the ramification locus. Propositions 4.5.3 and 4.5.4 imply that λ(z)=z𝜆𝑧𝑧\lambda(z)=z and the specialization of λ𝜆\lambda to k(z)𝑘𝑧k(z), denoted λk(z)subscript𝜆𝑘𝑧\lambda_{k(z)}, is the identity. Thus, the specialization of (A,τ)𝐴𝜏(A,\tau) to k(z)𝑘𝑧k(z), denoted (Ak(z),τk(z))subscript𝐴𝑘𝑧subscript𝜏𝑘𝑧(A_{k(z)},\tau_{k(z)}), is a central simple k(z)𝑘𝑧k(z)-algebra with an involution of the first kind.

Lemma 5.4.2.

With the above notation, the function fτ:Z{1,1}:subscript𝑓𝜏𝑍11f_{\tau}:Z\to\{1,-1\} determined by

fτ(z)={1 if τk(z) is orthogonal1 if τk(z) is symplecticsubscript𝑓𝜏𝑧cases1 if τk(z) is orthogonalotherwise1 if τk(z) is symplecticotherwisef_{\tau}(z)=\begin{cases}\phantom{-}1\quad\text{ if $\tau_{k(z)}$ is orthogonal}\\ -1\quad\text{ if $\tau_{k(z)}$ is symplectic}\end{cases}

is locally constant, and therefore determines a global section fτH0(Z,μ2,Z)subscript𝑓𝜏superscriptH0𝑍subscript𝜇2𝑍f_{\tau}\in\mathrm{H}^{0}(Z,\mu_{2,Z}).

Proof.

We may assume that degAdeg𝐴\operatorname{deg}A is constant; otherwise, decompose X𝑋X into a disjoint union of components on which this holds and work component-wise.

Let (AZ,τZ)subscript𝐴𝑍subscript𝜏𝑍(A_{Z},\tau_{Z}) denote the base change of (A,τ)𝐴𝜏(A,\tau) from X𝑋X to Z𝑍Z, namely (iA,iτ)i𝒪X(𝒪Z,λZ#=id𝒪Z)subscripttensor-productsuperscript𝑖subscript𝒪𝑋superscript𝑖𝐴superscript𝑖𝜏subscript𝒪𝑍superscriptsubscript𝜆𝑍#subscriptidsubscript𝒪𝑍(i^{*}A,i^{*}\tau)\otimes_{i^{*}{\mathcal{O}}_{X}}({\mathcal{O}}_{Z},\lambda_{Z}^{\#}=\mathrm{id}_{{\mathcal{O}}_{Z}}), where i:ZX:𝑖𝑍𝑋i:Z\to X denotes the inclusion map. For any point zZ𝑧𝑍z\in Z, the type of τ𝜏\tau at k(z)𝑘𝑧k(z) may be calculated relative to (AZ,τZ)subscript𝐴𝑍subscript𝜏𝑍(A_{Z},\tau_{Z}). We may therefore replace X𝑋X and (A,τ)𝐴𝜏(A,\tau) with Z𝑍Z and (AZ,τZ)subscript𝐴𝑍subscript𝜏𝑍(A_{Z},\tau_{Z}) to assume that Z=X𝑍𝑋Z=X and λ:XX:𝜆𝑋𝑋\lambda:X\to X is the trivial involution.

Let A+=ker(idAτ)subscript𝐴kernelsubscriptid𝐴𝜏A_{+}=\ker(\mathrm{id}_{A}-\tau) and A=ker(idA+τ)subscript𝐴kernelsubscriptid𝐴𝜏A_{-}=\ker(\mathrm{id}_{A}+\tau). That is, A+subscript𝐴A_{+} and Asubscript𝐴A_{-} are the sheaves and τ𝜏\tau-symmetric and τ𝜏\tau-antisymmetric elements in A𝐴A. Since λ=id𝜆id\lambda=\mathrm{id}, both A+subscript𝐴A_{+} and Asubscript𝐴A_{-} are 𝒪Xsubscript𝒪𝑋{\mathcal{O}}_{X}-modules, and since 222 is invertible on X𝑋X, the sequence 0AAid+τA+00subscript𝐴𝐴id𝜏subscript𝐴00\to A_{-}\hookrightarrow A\xrightarrow{\mathrm{id}+\tau}A_{+}\to 0 is split exact. Consequently, the sequence remains exact after base changing to k(z)𝑘𝑧k(z) for all zX𝑧𝑋z\in X, and so we may identify (A+)k(z)subscriptsubscript𝐴𝑘𝑧(A_{+})_{k(z)} with the τk(z)subscript𝜏𝑘𝑧\tau_{k(z)}-symmetric elements of Ak(z)subscript𝐴𝑘𝑧A_{k(z)}.

It is well known [knus_book_1998-1, §2A] that dim(A+)k(z)dimensionsubscriptsubscript𝐴𝑘𝑧\dim(A_{+})_{k(z)} equals 12n(n+1)12𝑛𝑛1\frac{1}{2}n(n+1) when τk(z)subscript𝜏𝑘𝑧\tau_{k(z)} is orthogonal and 12n(n1)12𝑛𝑛1\frac{1}{2}n(n-1) when τk(z)subscript𝜏𝑘𝑧\tau_{k(z)} is symplectic. Since A+subscript𝐴A_{+} is an 𝒪Xsubscript𝒪𝑋{\mathcal{O}}_{X}-summand of A𝐴A, it is locally free. Thus, the rank of A+subscript𝐴A_{+} is locally constant, and a fortiori so is fτsubscript𝑓𝜏f_{\tau}. ∎

We will prove, after a number of lemmas, that the element fτsubscript𝑓𝜏f_{\tau} determines the type of τ𝜏\tau. In the course of the proof, we shall see that the sheaf T𝑇T introduced in Subsection 5.2 is nothing but the pushforward to Y𝑌Y of the sheaf μ2,Wsubscript𝜇2𝑊\mu_{2,W} on W𝑊W.

Lemma 5.4.3.

Consider a commutative diagram

Xsuperscript𝑋\textstyle{X^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}u𝑢\scriptstyle{u}πsuperscript𝜋\scriptstyle{\pi^{\prime}}X𝑋\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}π𝜋\scriptstyle{\pi}Ysuperscript𝑌\textstyle{Y^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}v𝑣\scriptstyle{v}Y𝑌\textstyle{Y}

in which π:XY:𝜋superscript𝑋superscript𝑌\pi:X^{\prime}\to Y^{\prime} is a good C2subscript𝐶2C_{2}-quotient of schemes, resp. a C2subscript𝐶2C_{2}-quotient of Hausdorff topological spaces, and u𝑢u is C2subscript𝐶2C_{2}-equivariant. Let λsuperscript𝜆\lambda^{\prime} denote the involution of Xsuperscript𝑋X^{\prime} and let S,R,N,Tsuperscript𝑆superscript𝑅superscript𝑁superscript𝑇S^{\prime},R^{\prime},N^{\prime},T^{\prime} denote the sheaves corresponding to S,R,N,T𝑆𝑅𝑁𝑇S,R,N,T and constructed with π:XY:superscript𝜋superscript𝑋superscript𝑌\pi^{\prime}:X^{\prime}\to Y^{\prime} in place of π:XY:𝜋𝑋𝑌\pi:X\to Y. Then:

  1. (i)

    There are commutative squares of ring sheaves on Y𝑌Y and Ysuperscript𝑌Y^{\prime}, respectively:

    vRsubscript𝑣superscript𝑅\textstyle{v_{*}R^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}πu𝒪Xsubscript𝜋subscript𝑢subscript𝒪superscript𝑋\textstyle{\pi_{*}u_{*}{\mathcal{O}}_{X^{\prime}}}R𝑅\textstyle{R\ignorespaces\ignorespaces\ignorespaces\ignorespaces}πu#subscript𝜋subscript𝑢#\scriptstyle{\pi_{*}u_{\#}}vSsubscript𝑣superscript𝑆\textstyle{v_{*}S^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}vπ#subscript𝑣subscriptsuperscript𝜋#\scriptstyle{v_{*}\pi^{\prime}_{\#}}S𝑆\textstyle{S\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}v#subscript𝑣#\scriptstyle{v_{\#}}π#subscript𝜋#\scriptstyle{\pi_{\#}}      Rsuperscript𝑅\textstyle{R^{\prime}}vRsuperscript𝑣𝑅\textstyle{v^{*}R\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Ssuperscript𝑆\textstyle{S^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}π#subscriptsuperscript𝜋#\scriptstyle{\pi^{\prime}_{\#}}vSsuperscript𝑣𝑆\textstyle{v^{*}S\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}v#superscript𝑣#\scriptstyle{v^{\#}}vπ#superscript𝑣subscript𝜋#\scriptstyle{v^{*}\pi_{\#}}

    Here, the horizontal arrows of the right square are the adjoints of the horizontal arrows of the left square relative to the adjuntion between vsuperscript𝑣v^{*} and vsubscript𝑣v_{*}. Furthermore, in both squares, the top horizontal arrows are morphism of rings with involution.

  2. (ii)

    The left square of (i) induces morphisms NvN𝑁subscript𝑣superscript𝑁N\to v_{*}N^{\prime}, TvT𝑇subscript𝑣superscript𝑇T\to v_{*}T^{\prime} and H0(T)H0(T)superscriptH0𝑇superscriptH0superscript𝑇\mathrm{H}^{0}(T)\to\mathrm{H}^{0}(T^{\prime}). Furthermore, if (A,τ)𝐴𝜏(A,\tau) is an Azumaya 𝒪Xsubscript𝒪𝑋{\mathcal{O}}_{X}-algebra with a λ𝜆\lambda-involution and (A,τ)superscript𝐴superscript𝜏(A^{\prime},\tau^{\prime}) denotes the base change of (A,τ)𝐴𝜏(A,\tau) to Xsuperscript𝑋X^{\prime}, namely (uA,uτ)u𝒪X(𝒪X,λ)subscripttensor-productsuperscript𝑢subscript𝒪𝑋superscript𝑢𝐴superscript𝑢𝜏subscript𝒪superscript𝑋superscript𝜆(u^{*}A,u^{*}\tau)\otimes_{u^{*}{\mathcal{O}}_{X}}({\mathcal{O}}_{X^{\prime}},\lambda^{\prime}), then the image of ctπ(τ)H0(T)subscriptct𝜋𝜏superscriptH0𝑇\mathrm{ct}_{\pi}(\tau)\in\mathrm{H}^{0}(T) in H0(T)superscriptH0superscript𝑇\mathrm{H}^{0}(T^{\prime}) is ctπ(τ)subscriptctsuperscript𝜋superscript𝜏\mathrm{ct}_{\pi^{\prime}}(\tau^{\prime}).

Proof.

Part (i) and the first sentence of (ii) are straightforward from the definitions. We turn to prove the last statement of (ii).

We first claim that

(10) (πA,πτ)(vπA,vπτ)vR(R,λ).subscriptsuperscript𝜋superscript𝐴subscriptsuperscript𝜋superscript𝜏subscripttensor-productsuperscript𝑣𝑅superscript𝑣subscript𝜋𝐴superscript𝑣subscript𝜋𝜏superscript𝑅superscript𝜆(\pi^{\prime}_{*}A^{\prime},\pi^{\prime}_{*}\tau^{\prime})\cong(v^{*}\pi_{*}A,v^{*}\pi_{*}\tau)\otimes_{v^{*}R}(R^{\prime},\lambda^{\prime})\ .

To see this, observe that the relevant counit maps induce a ring homomorphism

π(vπAvRR)superscript𝜋subscripttensor-productsuperscript𝑣𝑅superscript𝑣subscript𝜋𝐴superscript𝑅\displaystyle\pi^{\prime*}(v^{*}\pi_{*}A\otimes_{v^{*}R}R^{\prime}) =πvπAπvπ𝒪Xππ𝒪Xabsentsubscripttensor-productsuperscript𝜋superscript𝑣subscript𝜋subscript𝒪𝑋superscript𝜋superscript𝑣subscript𝜋𝐴superscript𝜋subscriptsuperscript𝜋subscript𝒪superscript𝑋\displaystyle=\pi^{\prime*}v^{*}\pi_{*}A\otimes_{\pi^{\prime*}v^{*}\pi_{*}{\mathcal{O}}_{X}}\pi^{\prime*}\pi^{\prime}_{*}{\mathcal{O}}_{X^{\prime}}
=uππAuππ𝒪Xππ𝒪XuAu𝒪X𝒪X=Aabsentsubscripttensor-productsuperscript𝑢superscript𝜋subscript𝜋subscript𝒪𝑋superscript𝑢superscript𝜋subscript𝜋𝐴superscript𝜋subscriptsuperscript𝜋subscript𝒪superscript𝑋subscripttensor-productsuperscript𝑢subscript𝒪𝑋superscript𝑢𝐴subscript𝒪superscript𝑋superscript𝐴\displaystyle=u^{*}\pi^{*}\pi_{*}A\otimes_{u^{*}\pi^{*}\pi_{*}{\mathcal{O}}_{X}}\pi^{\prime*}\pi^{\prime}_{*}{\mathcal{O}}_{X^{\prime}}\to u^{*}A\otimes_{u^{*}{\mathcal{O}}_{X}}{\mathcal{O}}_{X^{\prime}}=A^{\prime}

which respects the relevant involutions. This morphism is adjoint to a morphism

(11) vπAvRRπAsubscripttensor-productsuperscript𝑣𝑅superscript𝑣subscript𝜋𝐴superscript𝑅subscriptsuperscript𝜋superscript𝐴v^{*}\pi_{*}A\otimes_{v^{*}R}R^{\prime}\to\pi^{\prime}_{*}A^{\prime}

which we claim to be the desired isomorphism. This is easy to see when A=Mn×n(𝒪X)𝐴subscriptM𝑛𝑛subscript𝒪𝑋A=\mathrm{M}_{n\times n}({\mathcal{O}}_{X}). In general, by Theorem 4.3.11, there exists a covering UY𝑈𝑌U\to Y such that A𝐴A becomes a matrix algebra after pulling back to XUsubscript𝑋𝑈X_{U}. Thus, (vπAvRR)YU(πA)YUsubscriptsubscripttensor-productsuperscript𝑣𝑅superscript𝑣subscript𝜋𝐴superscript𝑅subscriptsuperscript𝑌𝑈subscriptsubscriptsuperscript𝜋superscript𝐴subscriptsuperscript𝑌𝑈(v^{*}\pi_{*}A\otimes_{v^{*}R}R^{\prime})_{Y^{\prime}_{U}}\to(\pi^{\prime}_{*}A^{\prime})_{Y^{\prime}_{U}} is an isomorphism, and we conclude that so does (11).

With (10) at hand, let U,ψ,σ,g,h,ε𝑈𝜓𝜎𝑔𝜀U,\psi,\sigma,g,h,\varepsilon be as in Construction 5.2.5, applied to (A,τ)𝐴𝜏(A,\tau). We may assume that U𝑈U is represented by a covering of Y𝑌Y, denoted UY𝑈𝑌U\to Y. Let UYsuperscript𝑈superscript𝑌U^{\prime}\to Y^{\prime} be the pullback of UY𝑈𝑌U\to Y along v:YY:𝑣superscript𝑌𝑌v:Y^{\prime}\to Y, which corresponds to the sheaf vUsuperscript𝑣𝑈v^{*}U in 𝐘superscript𝐘{\mathbf{Y}}^{\prime}. Let ψ=vψvRUidRUsuperscript𝜓subscripttensor-productsuperscript𝑣subscript𝑅𝑈superscript𝑣𝜓subscriptidsubscriptsuperscript𝑅superscript𝑈\psi^{\prime}=v^{*}\psi\otimes_{v^{*}R_{U}}\mathrm{id}_{R^{\prime}_{U^{\prime}}} and let σ=ψτψ1=vσvRλsuperscript𝜎superscript𝜓superscript𝜏superscript𝜓1subscripttensor-productsuperscript𝑣𝑅superscript𝑣𝜎superscript𝜆\sigma^{\prime}=\psi^{\prime}\tau^{\prime}\psi^{\prime-1}=v^{*}\sigma\otimes_{v^{*}R}\lambda^{\prime}. The right square of (i) induces canonical maps vPGLn(R)=PGLn(vR)PGLn(R)superscript𝑣subscriptPGL𝑛𝑅subscriptPGL𝑛superscript𝑣𝑅subscriptPGL𝑛superscript𝑅v^{*}\operatorname{PGL}_{n}(R)=\operatorname{PGL}_{n}(v^{*}R)\to\operatorname{PGL}_{n}(R^{\prime}), vGLn(R)=GLn(vR)GLn(R)superscript𝑣subscriptGL𝑛𝑅subscriptGL𝑛superscript𝑣𝑅subscriptGL𝑛superscript𝑅v^{*}\operatorname{GL}_{n}(R)=\operatorname{GL}_{n}(v^{*}R)\to\operatorname{GL}_{n}(R^{\prime}) and vNNsuperscript𝑣𝑁superscript𝑁v^{*}N\to N^{\prime} (notice that vsuperscript𝑣v^{*} is exact). Let gsuperscript𝑔g^{\prime} be the image of vgvPGLn(R)(U)superscript𝑣𝑔superscript𝑣subscriptPGL𝑛𝑅superscript𝑈v^{*}g\in v^{*}\operatorname{PGL}_{n}(R)(U^{\prime}) in PGLn(R)(U)subscriptPGL𝑛superscript𝑅superscript𝑈\operatorname{PGL}_{n}(R^{\prime})(U^{\prime}), and define hGLn(R)(U)superscriptsubscriptGL𝑛superscript𝑅superscript𝑈h^{\prime}\in\operatorname{GL}_{n}(R^{\prime})(U^{\prime}) and εN(U)superscript𝜀superscript𝑁superscript𝑈\varepsilon^{\prime}\in N^{\prime}(U^{\prime}) similarly. It is easy to check that we can apply Construction 5.2.5 to (A,τ)superscript𝐴superscript𝜏(A^{\prime},\tau^{\prime}) using U,ψ,σ,g,h,εsuperscript𝑈superscript𝜓superscript𝜎superscript𝑔superscriptsuperscript𝜀U^{\prime},\psi^{\prime},\sigma^{\prime},g^{\prime},h^{\prime},\varepsilon^{\prime}. Consequently, the image of εsuperscript𝜀\varepsilon^{\prime} in T(U)superscript𝑇superscript𝑈T^{\prime}(U^{\prime}) agrees with the image of vεsuperscript𝑣𝜀v^{*}\varepsilon, which is exactly what we need to prove. ∎

Endowing Z𝑍Z with the trivial involution, we can apply Lemma 5.4.3 with the square

(12) Z𝑍\textstyle{Z\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}i𝑖\scriptstyle{i}πsuperscript𝜋\scriptstyle{\pi^{\prime}}X𝑋\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}π𝜋\scriptstyle{\pi}W𝑊\textstyle{W\ignorespaces\ignorespaces\ignorespaces\ignorespaces}j𝑗\scriptstyle{j}Y𝑌\textstyle{Y}

where πsuperscript𝜋\pi^{\prime} is the restriction of π𝜋\pi to Z𝑍Z. By Example 5.2.1, the sheaf Tsuperscript𝑇T^{\prime} is just μ2,Wsubscript𝜇2𝑊\mu_{2,W} and hence Lemma 5.4.3(ii) gives rise to a morphism

Ψ:Tjμ2,W.:Ψ𝑇subscript𝑗subscript𝜇2𝑊\Psi:T\to j_{*}\mu_{2,W}\ .
Lemma 5.4.4.

Ψ:Tjμ2,W:Ψ𝑇subscript𝑗subscript𝜇2𝑊\Psi:T\to j_{*}\mu_{2,W} is an isomorphism of abelian sheaves on Y𝑌Y.

Proof.

To show that ΨΨ\Psi is an isomorphism, it is enough to check the stalks. The topos-theoretic points of 𝐘𝐘{\mathbf{Y}} are recalled in the proofs of Corollaries 4.4.2 and 4.4.3; they are in correspondence with the set-theoretic points of Y𝑌Y.

Let p:𝐩𝐭𝐘:𝑝𝐩𝐭𝐘p:\mathbf{pt}\to{\mathbf{Y}} be a point, corresponding to yY𝑦𝑌y\in Y. Since psuperscript𝑝p^{*} is exact, pNsuperscript𝑝𝑁p^{*}N is the kernel of xxλx:pR×pR×:maps-to𝑥superscript𝑥𝜆𝑥superscript𝑝superscript𝑅superscript𝑝superscript𝑅x\mapsto x^{\lambda}x:{p^{*}R^{\times}}\to{p^{*}R^{\times}} and pTsuperscript𝑝𝑇p^{*}T is the cokernel of xx1xλ:pR×pN:maps-to𝑥superscript𝑥1superscript𝑥𝜆superscript𝑝superscript𝑅superscript𝑝𝑁x\mapsto x^{-1}x^{\lambda}:{p^{*}R^{\times}}\to p^{*}N.

Suppose that yW𝑦𝑊y\notin W. Then, since j:WY:𝑗𝑊𝑌j:W\to Y is a closed embedding, pjμ2,W=0superscript𝑝subscript𝑗subscript𝜇2𝑊0p^{*}j_{*}\mu_{2,W}=0. On the other hand, since π𝜋\pi is unramified at y𝑦y, it is unramified at a neighborhood of y𝑦y and hence pT=0superscript𝑝𝑇0p^{*}T=0 by Proposition 5.2.2. Thus, pΨ:pTpjμ2,W:superscript𝑝Ψsuperscript𝑝𝑇superscript𝑝subscript𝑗subscript𝜇2𝑊p^{*}\Psi:p^{*}T\to p^{*}j_{*}\mu_{2,W} is an isomorphism.

Suppose henceforth that yW𝑦𝑊y\in W. Then π𝜋\pi is ramified at y𝑦y. We claim that pRsuperscript𝑝𝑅p^{*}R is local and λ𝜆\lambda induces the identity map on its residue field. This is evident from the definitions in the topological case, see Proposition 4.5.4. In the scheme-theoertic case, this follows from condition (c) in Proposition 4.5.3 and Theorem 3.3.8 after noting that SpecpR=X×YSpec𝒪Y,yshSpecsuperscript𝑝𝑅subscript𝑌𝑋Specsuperscriptsubscript𝒪𝑌𝑦sh\operatorname{Spec}p^{*}R=X\times_{Y}\operatorname{Spec}{\mathcal{O}}_{Y,y}^{\mathrm{sh}}.

Now, we have pjμ2,W={±1}superscript𝑝subscript𝑗subscript𝜇2𝑊plus-or-minus1p^{*}j_{*}\mu_{2,W}=\{\pm 1\}. With the notation of Lemma 5.4.3, applied to the square (12), the morphism NjN𝑁subscript𝑗superscript𝑁N\to j_{*}N^{\prime} is just a restriction of the morphism RjR=j𝒪W𝑅subscript𝑗superscript𝑅subscript𝑗subscript𝒪𝑊R\to j_{*}R^{\prime}=j_{*}{\mathcal{O}}_{W}. This implies that the images of 1,1pN11superscript𝑝𝑁-1,1\in p^{*}N in pTsuperscript𝑝𝑇p^{*}T are mapped under pΨsuperscript𝑝Ψp^{*}\Psi to 1,1pjμ2,W11superscript𝑝subscript𝑗subscript𝜇2𝑊-1,1\in p^{*}j_{*}\mu_{2,W}, respectively, so pΨsuperscript𝑝Ψp^{*}\Psi is surjective.

To finish, we show that pTsuperscript𝑝𝑇p^{*}T consists of at most 222 elements. Every tpT𝑡superscript𝑝𝑇t\in p^{*}T is represented by some εpN𝜀superscript𝑝𝑁\varepsilon\in p^{*}N. Since 2pR×2superscript𝑝superscript𝑅2\in{p^{*}R^{\times}} and pRsuperscript𝑝𝑅p^{*}R is local, either 1+ε1𝜀1+\varepsilon or 1ε1𝜀1-\varepsilon is invertible. Suppose β:=1+εpR×assign𝛽1𝜀superscript𝑝superscript𝑅\beta:=1+\varepsilon\in{p^{*}R^{\times}}. Since ελ=ε1superscript𝜀𝜆superscript𝜀1\varepsilon^{\lambda}=\varepsilon^{-1}, we have εβλ=β𝜀superscript𝛽𝜆𝛽\varepsilon\beta^{\lambda}=\beta, or rather, ε=(β1)λβ𝜀superscriptsuperscript𝛽1𝜆𝛽\varepsilon=(\beta^{-1})^{\lambda}\beta, which implies t=1¯𝑡¯1t=\overline{1}. Similarly, when 1εpR×1𝜀superscript𝑝superscript𝑅1-\varepsilon\in{p^{*}R^{\times}}, we find that t=1¯𝑡¯1{t}=\overline{-1}. It follows that pT={1¯,1¯}superscript𝑝𝑇¯1¯1p^{*}T=\{\overline{1},\overline{-1}\} and the proof is complete. ∎

We finally prove the main result of this subsection.

Theorem 5.4.5.

With Notation 5.4.1, Let (A,τ)𝐴𝜏(A,\tau) and (A,τ)superscript𝐴superscript𝜏(A^{\prime},\tau^{\prime}) be Azumaya 𝒪Xsubscript𝒪𝑋{\mathcal{O}}_{X}-algebras with λ𝜆\lambda-involutions, and let fτ,fτH0(Z,μ2,Z)subscript𝑓𝜏subscript𝑓superscript𝜏superscriptH0𝑍subscript𝜇2𝑍f_{\tau},f_{\tau^{\prime}}\in\mathrm{H}^{0}(Z,\mu_{2,Z}) be as in Lemma 5.4.2. Then:

  1. (i)

    τ𝜏\tau and τsuperscript𝜏\tau^{\prime} have the same type if and only if fτ=fτsubscript𝑓𝜏subscriptsuperscript𝑓𝜏f_{\tau}=f^{\prime}_{\tau}.

  2. (ii)

    There is a group isomorphism Φ:H0(T)H0(Z,μ2,Z):ΦsuperscriptH0𝑇superscriptH0𝑍subscript𝜇2𝑍\Phi:\mathrm{H}^{0}(T)\to\mathrm{H}^{0}(Z,\mu_{2,Z}) such that Φ(ctπ(τ))=fτΦsubscriptct𝜋𝜏subscript𝑓𝜏\Phi(\mathrm{ct}_{\pi}(\tau))=f_{\tau} for all (A,τ)𝐴𝜏(A,\tau).

Remark 5.4.6.

We do not know whether every fH0(Z,μ2,Z)𝑓superscriptH0𝑍subscript𝜇2𝑍f\in\mathrm{H}^{0}(Z,\mu_{2,Z}) arises as fτsubscript𝑓𝜏f_{\tau} for some (A,τ)𝐴𝜏(A,\tau), see Remark 5.2.16.

Proof.

By Theorem 5.2.13 and Example 5.2.12, in order prove (i), it is enough to prove that ctπ(τ)=ctπ(τ)subscriptct𝜋𝜏subscriptct𝜋superscript𝜏\mathrm{ct}_{\pi}(\tau)=\mathrm{ct}_{\pi}(\tau^{\prime}), and this follows if we prove (ii).

Apply Lemma 5.4.3 and its notation to the square (12). The lemma gives rise to a morphism of sheaves TjT=jμ2,W𝑇subscript𝑗superscript𝑇subscript𝑗subscript𝜇2𝑊T\to j_{*}T^{\prime}=j_{*}\mu_{2,W}, which is an isomorphism by Lemma 5.4.4. This in turn induces an isomorphism

H0(Y,T)H0(Y,jT)=H0(W,T),superscriptH0𝑌𝑇superscriptH0𝑌subscript𝑗superscript𝑇superscriptH0𝑊superscript𝑇\mathrm{H}^{0}(Y,T)\to\mathrm{H}^{0}(Y,j_{*}T^{\prime})=\mathrm{H}^{0}(W,T^{\prime})\ ,

such that ctπ(A,τ)subscriptct𝜋𝐴𝜏\mathrm{ct}_{\pi}(A,\tau) is mapped to ctπ(AZ,τZ)subscriptctsuperscript𝜋subscript𝐴𝑍subscript𝜏𝑍\mathrm{ct}_{\pi^{\prime}}(A_{Z},\tau_{Z}), where (AZ,τZ)subscript𝐴𝑍subscript𝜏𝑍(A_{Z},\tau_{Z}) denotes the base change of (A,τ)𝐴𝜏(A,\tau) to Z𝑍Z. Since π:ZW:superscript𝜋𝑍𝑊\pi^{\prime}:Z\to W is an isomorphism, this gives rise to an isomorphism

H0(Y,T)H0(W,T)=H0(W,μ2,W)H0(Z,μ2,Z),superscriptH0𝑌𝑇superscriptH0𝑊superscript𝑇superscriptH0𝑊subscript𝜇2𝑊superscriptH0𝑍subscript𝜇2𝑍\mathrm{H}^{0}(Y,T)\to\mathrm{H}^{0}(W,T^{\prime})=\mathrm{H}^{0}(W,\mu_{2,W})\cong\mathrm{H}^{0}(Z,\mu_{2,Z})\ ,

which we take to be ΦΦ\Phi. It remains to show that Φ(ctπ(A,τ))=fτΦsubscriptct𝜋𝐴𝜏subscript𝑓𝜏\Phi(\mathrm{ct}_{\pi}(A,\tau))=f_{\tau}.

Since fτZ=fτsubscript𝑓subscript𝜏𝑍subscript𝑓𝜏f_{\tau_{Z}}=f_{\tau}, and since the image of ctπ(A,τ)subscriptct𝜋𝐴𝜏\mathrm{ct}_{\pi}(A,\tau) in H0(T)superscriptH0superscript𝑇\mathrm{H}^{0}(T^{\prime}) is ctπ(AZ,τZ)subscriptctsuperscript𝜋subscript𝐴𝑍subscript𝜏𝑍\mathrm{ct}_{\pi^{\prime}}(A_{Z},\tau_{Z}), it is enough to show that the image of ctπ(AZ,τZ)H0(W,T)subscriptctsuperscript𝜋subscript𝐴𝑍subscript𝜏𝑍superscriptH0𝑊superscript𝑇\mathrm{ct}_{\pi^{\prime}}(A_{Z},\tau_{Z})\in\mathrm{H}^{0}(W,T^{\prime}) in H0(Z,μ2,Z)superscriptH0𝑍subscript𝜇2𝑍\mathrm{H}^{0}(Z,\mu_{2,Z}) is fτZsubscript𝑓subscript𝜏𝑍f_{\tau_{Z}}. To this end, we replace π:XY:𝜋𝑋𝑌\pi:X\to Y and (A,τ)𝐴𝜏(A,\tau) with π:ZW:superscript𝜋𝑍𝑊\pi^{\prime}:Z\to W and (AZ,τZ)subscript𝐴𝑍subscript𝜏𝑍(A_{Z},\tau_{Z}). Now, λ𝜆\lambda is the trivial involution and we may assume that Y=X𝑌𝑋Y=X and π𝜋\pi is the identity map. The map H0(W,T)H0(Z,μ2,Z)superscriptH0𝑊superscript𝑇superscriptH0𝑍subscript𝜇2𝑍\mathrm{H}^{0}(W,T^{\prime})\to\mathrm{H}^{0}(Z,\mu_{2,Z}) is just the identity map H0(X,μ2,X)H0(X,μ2,X)superscriptH0𝑋subscript𝜇2𝑋superscriptH0𝑋subscript𝜇2𝑋\mathrm{H}^{0}(X,\mu_{2,X})\to\mathrm{H}^{0}(X,\mu_{2,X}), see Example 5.2.1, and the proof reduces to showing that ct(τ)=fτct𝜏subscript𝑓𝜏\mathrm{ct}(\tau)=f_{\tau}.

Let xX𝑥𝑋x\in X, and let U,σ,h,ε𝑈𝜎𝜀U,\sigma,h,\varepsilon be as in Construction 5.2.5, applied to (A,τ)𝐴𝜏(A,\tau). We may assume that the sheaf U𝑈U in 𝐘=𝐗𝐘𝐗{\mathbf{Y}}={\mathbf{X}} is represented by a covering UX𝑈𝑋U\to X. Since λ𝜆\lambda is the trivial involution, εN(U)=μ2,X(U)𝜀𝑁𝑈subscript𝜇2𝑋𝑈\varepsilon\in N(U)=\mu_{2,X}(U), and since μ2,Xsubscript𝜇2𝑋\mu_{2,X} is the constant sheaf {±1}plus-or-minus1\{\pm 1\} on X𝑋X, there is a covering U1U1Usquare-unionsubscript𝑈1subscript𝑈1𝑈U_{1}\sqcup U_{-1}\to U such that ε|U1=1evaluated-at𝜀subscript𝑈11\varepsilon|_{U_{-1}}=-1 and ε|U1=1evaluated-at𝜀subscript𝑈11\varepsilon|_{U_{1}}=1.

There is c{±1}𝑐plus-or-minus1c\in\{\pm 1\} and uUc𝑢subscript𝑈𝑐u\in U_{c} such that x𝑥x is the image of u𝑢u under UcXsubscript𝑈𝑐𝑋U_{c}\to X. It is immediate from the definition of t:=ct(τ)assign𝑡ct𝜏t:=\mathrm{ct}(\tau) that t(x)=c𝑡𝑥𝑐t(x)=c. Let k(u)𝑘𝑢k(u) denote the residue field of u𝑢u. By construction, (AUc,τUc)(Mn×n(𝒪X),σ)subscript𝐴subscript𝑈𝑐subscript𝜏subscript𝑈𝑐subscriptM𝑛𝑛subscript𝒪𝑋𝜎(A_{U_{c}},\tau_{U_{c}})\cong(\mathrm{M}_{n\times n}({\mathcal{O}}_{X}),\sigma), where σ𝜎\sigma is given section-wise by x(hxh1)trmaps-to𝑥superscript𝑥superscript1trx\mapsto(hxh^{-1})^{\text{\rm tr}} and chtr=h𝑐superscripttrch^{\text{\rm tr}}=h. Thus, τUcsubscript𝜏subscript𝑈𝑐\tau_{U_{c}} is orthogonal when c=1𝑐1c=1 and symplectic when c=1𝑐1c=-1. The same applies to τk(u):Ak(u)Ak(u):subscript𝜏𝑘𝑢subscript𝐴𝑘𝑢subscript𝐴𝑘𝑢\tau_{k(u)}:A_{k(u)}\to A_{k(u)}. Since (Ak(u),τk(u))=(Ak(x),τk(x))k(x)(k(u),id)subscript𝐴𝑘𝑢subscript𝜏𝑘𝑢subscripttensor-product𝑘𝑥subscript𝐴𝑘𝑥subscript𝜏𝑘𝑥𝑘𝑢id(A_{k(u)},\tau_{k(u)})=(A_{k(x)},\tau_{k(x)})\otimes_{k(x)}(k(u),\mathrm{id}), it follows that τk(x):Ak(x)Ak(x):subscript𝜏𝑘𝑥subscript𝐴𝑘𝑥subscript𝐴𝑘𝑥\tau_{k(x)}:A_{k(x)}\to A_{k(x)} is orthogonal when t(x)=1𝑡𝑥1t(x)=1 and symplectic when t(x)=1𝑡𝑥1t(x)=-1. This means t=fτ𝑡subscript𝑓𝜏t=f_{\tau}, so we are done. ∎

6. Brauer Classes Supporting an Involution

6.1. Introduction

Let K𝐾K be a field and let λ:KK:𝜆𝐾𝐾\lambda:K\to K be an involution with fixed field F𝐹F. The central simple K𝐾K-algebras admitting a λ𝜆\lambda-involution were characterized by Albert, Riehm and Scharlau, see for instance [knus_book_1998-1, Thm. 3.1], who proved:

Theorem.

Let A𝐴A be a central simple K𝐾K-algebra. Then:

  1. (i)

    (Albert) When λ=id𝜆id\lambda=\mathrm{id}, A𝐴A admits a λ𝜆\lambda-involution if and only if 2[A]=02delimited-[]𝐴02[A]=0 in Br(K)Br𝐾\operatorname{Br}(K).

  2. (ii)

    (Albert–Riehm–Scharlau) When λid𝜆id\lambda\neq\mathrm{id}, A𝐴A admits a λ𝜆\lambda-involution if and only if [coresK/F(A)]=0delimited-[]subscriptcores𝐾𝐹𝐴0[\operatorname{cores}_{K/F}(A)]=0 in Br(F)Br𝐹\operatorname{Br}(F).

Here, coresK/F(A)subscriptcores𝐾𝐹𝐴\operatorname{cores}_{K/F}(A) is the corestriction algebra of A𝐴A, whose definition we recall below.

The Albert–Riehm–Scharlau Theorem does not, in general, hold if we replace K𝐾K with an arbitrary ring. However, in [saltman_azumaya_1978], Saltman showed that the Brauer classes admitting a representative with a λ𝜆\lambda-involution can still be characterized similarly.

Theorem (Saltman [saltman_azumaya_1978, Th. 3.1]).

Let R𝑅R be a ring, let λ:RR:𝜆𝑅𝑅\lambda:R\to R be an involution and let S𝑆S be the fixed ring of λ𝜆\lambda. Let A𝐴A be an Azumaya R𝑅R-algebra. Then:

  1. (i)

    When λ=id𝜆id\lambda=\mathrm{id}, there exists A[A]superscript𝐴delimited-[]𝐴A^{\prime}\in[A] such that Asuperscript𝐴A^{\prime} admits a λ𝜆\lambda-involution if and only if 2[A]=02delimited-[]𝐴02[A]=0 in Br(R)Br𝑅\operatorname{Br}(R).

  2. (ii)

    When R𝑅R is quadratic étale over S𝑆S, there exists A[A]superscript𝐴delimited-[]𝐴A^{\prime}\in[A] such that Asuperscript𝐴A^{\prime} admits a λ𝜆\lambda-involution if and only if [coresR/S(A)]=0delimited-[]subscriptcores𝑅𝑆𝐴0[\operatorname{cores}_{R/S}(A)]=0 in Br(S)Br𝑆\operatorname{Br}(S).

A later proof by Knus, Parimala and Srinivas [knus_azumaya_1990, Thms. 4.1, 4.2] applies in the generality of schemes and also implies that the representative Asuperscript𝐴A^{\prime} can be chosen such that degA=2degAdegsuperscript𝐴2deg𝐴\operatorname{deg}A^{\prime}=2\operatorname{deg}A.

In this section, we extend Saltman’s theorem to locally ringed topoi with involution. We note that our generalization implies in particular that Salman’s theorem applies to topological Azumaya algebras. Furthermore, while Saltman’s theorem assumes that λ=id𝜆id\lambda=\mathrm{id}, or R𝑅R is quadratic étale over the fixed ring of λ𝜆\lambda, our result will apply without any restriction on the involution. Finally, we also characterize the possible types, or more precisely, coarse types, of the involutions of the various representatives A[A]superscript𝐴delimited-[]𝐴A^{\prime}\in[A].

Notation 6.1.1.

Throughout this section, let 𝐗𝐗{\mathbf{X}} be a locally ringed topos with ring object 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}} and involution λ=(Λ,ν,λ)𝜆Λ𝜈𝜆\lambda=(\Lambda,\nu,\lambda), and let π:𝐗𝐘:𝜋𝐗𝐘\pi:{\mathbf{X}}\to{\mathbf{Y}} be an exact quotient relative to λ𝜆\lambda, see 4.3. Recall that such quotients arise, for instance, from C2subscript𝐶2C_{2}-quotients of schemes or Hausdorff topological spaces as explained in Examples 4.3.3 and 4.3.4. In such cases, we shall work with the original schemes, resp. topological spaces, denoted X𝑋X and Y𝑌Y, rather than the associated ringed topoi.

As in Section 5, we write S=𝒪𝐘𝑆subscript𝒪𝐘S={\mathcal{O}}_{\mathbf{Y}} and R=π𝒪𝐗𝑅subscript𝜋subscript𝒪𝐗R=\pi_{*}{\mathcal{O}}_{\mathbf{X}}.

We sometimes omit bases when evaluating cohomology; the base will always be clear from the context. If A𝐴A is an abelian group in 𝐗𝐗{\mathbf{X}}, we shall freely identify Hi(𝐗,A)superscriptH𝑖𝐗𝐴\mathrm{H}^{i}(\mathbf{X},A), written Hi(A)superscriptH𝑖𝐴\mathrm{H}^{i}(A), with Hi(𝐘,πA)superscriptH𝑖𝐘subscript𝜋𝐴\mathrm{H}^{i}(\mathbf{Y},\pi_{*}A), written Hi(πA)superscriptH𝑖subscript𝜋𝐴\mathrm{H}^{i}(\pi_{*}A), using Theorem 4.3.6.

6.2. The Cohomological Transfer Map

The corestriction map considered in the aforementioned theorems of Albert–Riehm–Scharlau and Saltman is a special case of the cohomological transfer map, which will feature in our generalization of Saltman’s theorem.

Definition 6.2.1.

The cohomological λ𝜆\lambda-transfer map transfλ:H2(𝐗,𝒪𝐗×)H2(𝐘,𝒪𝐘×):subscripttransf𝜆superscriptH2𝐗superscriptsubscript𝒪𝐗superscriptH2𝐘superscriptsubscript𝒪𝐘\operatorname{transf}_{\lambda}:\mathrm{H}^{2}(\mathbf{X},{\mathcal{O}_{\mathbf{X}}^{\times}})\to\mathrm{H}^{2}(\mathbf{Y},{{\mathcal{O}}_{\mathbf{Y}}^{\times}}) is the composite of the isomorphism H2(𝐗,𝒪𝐗×)H2(𝐘,π𝒪𝐗×)similar-tosuperscriptH2𝐗superscriptsubscript𝒪𝐗superscriptH2𝐘subscript𝜋superscriptsubscript𝒪𝐗\mathrm{H}^{2}({\mathbf{X}},{{\mathcal{O}}_{\mathbf{X}}^{\times}})\xrightarrow{\sim}\mathrm{H}^{2}({\mathbf{Y}},{\pi_{*}{\mathcal{O}}_{\mathbf{X}}^{\times}}) induced by πsubscript𝜋\pi_{*}, see Theorem 4.3.6, and the morphism H2(𝐘,π𝒪𝐗×)H2(𝐘,𝒪𝐘×)superscriptH2𝐘subscript𝜋superscriptsubscript𝒪𝐗superscriptH2𝐘superscriptsubscript𝒪𝐘\mathrm{H}^{2}({\mathbf{Y}},{\pi_{*}{\mathcal{O}}_{\mathbf{X}}^{\times}})\to\mathrm{H}^{2}({\mathbf{Y}},{{\mathcal{O}}_{\mathbf{Y}}^{\times}}) induced by the λ𝜆\lambda-norm map xxλx:π𝒪𝐗×𝒪𝐘×:maps-to𝑥superscript𝑥𝜆𝑥subscript𝜋superscriptsubscript𝒪𝐗superscriptsubscript𝒪𝐘x\mapsto x^{\lambda}x:{\pi_{*}{\mathcal{O}}_{\mathbf{X}}^{\times}}\to{{\mathcal{O}}_{\mathbf{Y}}^{\times}}. When no confusion can arise, we shall omit λ𝜆\lambda, simply writing transftransf\operatorname{transf} for transfλsubscripttransf𝜆\operatorname{transf}_{\lambda}, and calling it the transfer map.

Example 6.2.2.

If the involution λ𝜆\lambda of 𝐗𝐗{\mathbf{X}} is weakly trivial and π:𝐗𝐘:𝜋𝐗𝐘\pi:{\mathbf{X}}\to{\mathbf{Y}} is the trivial quotient, see Example 4.3.5, then the λ𝜆\lambda-norm is the squaring map xx2:π𝒪𝐗×π𝒪𝐗×=𝒪𝐘×:maps-to𝑥superscript𝑥2subscript𝜋superscriptsubscript𝒪𝐗subscript𝜋superscriptsubscript𝒪𝐗superscriptsubscript𝒪𝐘x\mapsto x^{2}:{\pi_{*}{\mathcal{O}}_{\mathbf{X}}^{\times}}\to{\pi_{*}{\mathcal{O}}_{\mathbf{X}}^{\times}}={{\mathcal{O}}_{\mathbf{Y}}^{\times}}, and so transfλ:H2(𝒪𝐗×)H2(𝒪𝐘×)H2(𝒪𝐗×):subscripttransf𝜆superscriptH2superscriptsubscript𝒪𝐗superscriptH2superscriptsubscript𝒪𝐘superscriptH2superscriptsubscript𝒪𝐗\operatorname{transf}_{\lambda}:\mathrm{H}^{2}(\mathcal{O}_{\mathbf{X}}^{\times})\to\mathrm{H}^{2}(\mathcal{O}_{\mathbf{Y}}^{\times})\cong\mathrm{H}^{2}({{\mathcal{O}}_{\mathbf{X}}^{\times}}) is multiplication by 222.

Example 6.2.3.

Let π:XY:𝜋𝑋𝑌\pi:X\to Y be a quadratic étale morphism of schemes, and let λ:XX:𝜆𝑋𝑋\lambda:X\to X be the canonical Y𝑌Y-involution of X𝑋X, given section-wise by xλ=TrX/Y(x)xsuperscript𝑥𝜆subscriptTr𝑋𝑌𝑥𝑥x^{\lambda}=\mathrm{Tr}_{X/Y}(x)-x. We consider the exact quotient obtained from π𝜋\pi and λ𝜆\lambda by taking étale ringed topoi, see Example 4.3.3. In this case, the transfer map transfλ:Hét2(X,𝒪X×)Hét2(Y,𝒪Y×):subscripttransf𝜆subscriptsuperscriptH2ét𝑋superscriptsubscript𝒪𝑋subscriptsuperscriptH2ét𝑌superscriptsubscript𝒪𝑌\operatorname{transf}_{\lambda}:\mathrm{H}^{2}_{\text{\'{e}t}}(X,{{\mathcal{O}}_{X}^{\times}})\to\mathrm{H}^{2}_{\text{\'{e}t}}(Y,{{\mathcal{O}}_{Y}^{\times}}) is, by definition, the corestriction map coresX/Y:Hét2(X,𝒪X×)Hét2(Y,𝒪Y×):subscriptcores𝑋𝑌subscriptsuperscriptH2ét𝑋superscriptsubscript𝒪𝑋subscriptsuperscriptH2ét𝑌superscriptsubscript𝒪𝑌\operatorname{cores}_{X/Y}:\mathrm{H}^{2}_{\text{\'{e}t}}(X,{{\mathcal{O}}_{X}^{\times}})\to\mathrm{H}^{2}_{\text{\'{e}t}}(Y,{{\mathcal{O}}_{Y}^{\times}}). Moreover, coresX/Ysubscriptcores𝑋𝑌\operatorname{cores}_{X/Y} restricts to a map coresX/Y:Br(X)Br(Y):subscriptcores𝑋𝑌Br𝑋Br𝑌\operatorname{cores}_{X/Y}:\operatorname{Br}(X)\to\operatorname{Br}(Y) which can be described explicitly on the level of Azumaya algebras: Let A𝐴A be an Azumaya 𝒪Xsubscript𝒪𝑋{\mathcal{O}}_{X}-algebra. The corestriction algebra coresX/Y(A)subscriptcores𝑋𝑌𝐴\operatorname{cores}_{X/Y}(A) is an Azumaya 𝒪Ysubscript𝒪𝑌{\mathcal{O}}_{Y}-algebra defined as the 𝒪Ysubscript𝒪𝑌{\mathcal{O}}_{Y}-subalgebra of π(A𝒪XλA)subscript𝜋subscripttensor-productsubscript𝒪𝑋𝐴superscript𝜆𝐴\pi_{*}(A\otimes_{{\mathcal{O}}_{X}}\lambda^{*}A) fixed by the exchange automorphism, given by xyyxmaps-totensor-product𝑥𝑦tensor-product𝑦𝑥x\otimes y\mapsto y\otimes x on sections. The map coresX/Y:Br(X)Br(Y):subscriptcores𝑋𝑌Br𝑋Br𝑌\operatorname{cores}_{X/Y}:\operatorname{Br}(X)\to\operatorname{Br}(Y) is then given by [A][coresX/Y(A)]maps-todelimited-[]𝐴delimited-[]subscriptcores𝑋𝑌𝐴[A]\mapsto[\operatorname{cores}_{X/Y}(A)], see [knus_azumaya_1990, p. 68] (the diagram on that page contains a misprint, on the right column, both ‘S𝑆S’s should be ‘R𝑅R’s).

Remark 6.2.4.

In contrast to the situation in Examples 6.2.2 and 6.2.3, we do not know whether

transfλ:H2(𝐗,𝒪𝐗×)H2(𝐘,𝒪𝐘×):subscripttransf𝜆superscriptH2𝐗superscriptsubscript𝒪𝐗superscriptH2𝐘superscriptsubscript𝒪𝐘\operatorname{transf}_{\lambda}:\mathrm{H}^{2}(\mathbf{X},{\mathcal{O}_{\mathbf{X}}^{\times}})\to\mathrm{H}^{2}(\mathbf{Y},{{\mathcal{O}}_{\mathbf{Y}}^{\times}})

restricts to a map between the Brauer groups Br(𝐗,𝒪𝐗)Br(𝐘,𝒪𝐘)Br𝐗subscript𝒪𝐗Br𝐘subscript𝒪𝐘\operatorname{Br}({\mathbf{X}},{\mathcal{O}}_{\mathbf{X}})\to\operatorname{Br}({\mathbf{Y}},{\mathcal{O}}_{\mathbf{Y}}), even in the cases induced by a good C2subscript𝐶2C_{2}-quotient of schemes π:XY:𝜋𝑋𝑌\pi:X\to Y. Some positive results appear in [auel_parimala_suresh_2015, Lem. 5.1, Rmk. 5.2]. Also, when R𝑅R is locally free of rank 222 over S𝑆S, Ferrand [ferrand_1998_norm_functors] constructs a universal norm functor taking R𝑅R-algebras to S𝑆S-algebras, which coincides with coresR/Ssubscriptcores𝑅𝑆\operatorname{cores}_{R/S} when R𝑅R is quadratic étale over S𝑆S, but it is a priori not clear whether it takes Azumaya R𝑅R-algebras to Azumaya S𝑆S-algebras in general. We hope to address this problem in a subsequent work.

We further note that without assuming that π𝜋\pi is unramified, the construction of Example 6.2.3 may produce an algebra which is not Azumaya. For example, it can be checked directly that coresR/S(M2×2(R))subscriptcores𝑅𝑆subscriptM22𝑅\operatorname{cores}_{R/S}(\mathrm{M}_{2\times 2}(R)) is not Azumaya over S𝑆S when S=𝑆S=\mathbb{C}, R=[x]/(x2)𝑅delimited-[]𝑥superscript𝑥2R=\mathbb{C}[x]/(x^{2}), and λ:RR:𝜆𝑅𝑅\lambda:R\to R is the \mathbb{C}-involution taking x𝑥x to x𝑥-x.

Example 6.2.5.

In the case where X𝑋X is a Hausdorff topological space with a free C2subscript𝐶2C_{2}-action and π:XY:=X/C2:𝜋𝑋𝑌assign𝑋subscript𝐶2\pi:X\to Y:=X/C_{2} is the corresponding 222-sheeted covering, the construction

transf:H2(X,S1)H2(X,𝒪X×)H2(Y,𝒪Y×)H2(Y,S1):transfsuperscriptH2𝑋superscript𝑆1superscriptH2𝑋superscriptsubscript𝒪𝑋superscriptH2𝑌superscriptsubscript𝒪𝑌superscriptH2𝑌superscript𝑆1\operatorname{transf}:\mathrm{H}^{2}(X,S^{1})\cong\mathrm{H}^{2}(X,\mathcal{O}_{X}^{\times})\to\mathrm{H}^{2}(Y,\mathcal{O}_{Y}^{\times})\cong\mathrm{H}^{2}(Y,S^{1})

is a special case of the usual transfer map for a 222-sheeted cover. This can be proved by considering transftransf\operatorname{transf} on the level of 222-cocycles. See also [piacenza_transfer_1984, Sec. 3.3] and note that πsuperscript𝜋\pi^{*} takes 𝒪Y×superscriptsubscript𝒪𝑌{\mathcal{O}_{Y}^{\times}}, the sheaf of nonvanishing continuous complex-valued functions on Y𝑌Y, to 𝒪Xsubscript𝒪𝑋\mathcal{O}_{X} on 𝐗𝐗\mathbf{X}.

Remark 6.2.6.

There is a notion of transfer for ramified covers XX/G𝑋𝑋𝐺X\to X/G where G𝐺G is a finite group, in particular, when G=C2𝐺subscript𝐶2G=C_{2}. This may be found in [aguilar_transfer_2010]. It seems likely, that map transfλsubscripttransf𝜆\operatorname{transf}_{\lambda} given here is a special case of that construction, but we do not pursue this further.

6.3. Brauer Classes Supporting a λ𝜆\lambda-Involution

In this subsection, we characterize those Brauer classes in Br(𝐗,𝒪𝐗)Br𝐗subscript𝒪𝐗\operatorname{Br}({\mathbf{X}},{\mathcal{O}}_{\mathbf{X}}) admitting a representative with a λ𝜆\lambda-involution, thus generalizing Saltman’s theorem [saltman_azumaya_1978, Th. 3.1].

We remind the reader that the notational conventions of Notation 6.1.1 are still in effect. In particular, S:=𝒪𝐘assign𝑆subscript𝒪𝐘S:={\mathcal{O}}_{\mathbf{Y}} is a local ring object in 𝐘𝐘{\mathbf{Y}} and R:=π𝒪𝐗assign𝑅subscript𝜋subscript𝒪𝐗R:=\pi_{*}{\mathcal{O}}_{\mathbf{X}} is a commutative S𝑆S-algebra with involution λ𝜆\lambda such that the fixed ring of λ𝜆\lambda is S𝑆S.

As in Subsection 5.2, we define N𝑁N to be the kernel of the λ𝜆\lambda-norm xxλx:R×S×:maps-to𝑥superscript𝑥𝜆𝑥superscript𝑅superscript𝑆x\mapsto x^{\lambda}x:{R^{\times}}\to{S^{\times}} and let T𝑇T be the quotient of N𝑁N by the image of the map xxλx1:R×N:maps-to𝑥superscript𝑥𝜆superscript𝑥1superscript𝑅𝑁x\mapsto x^{\lambda}x^{-1}:R^{\times}\to N. Recall that cTyp(λ):=H0(T)assigncTyp𝜆superscriptH0𝑇{\mathrm{cTyp}({\lambda})}:=\mathrm{H}^{0}(T) is the group of coarse λ𝜆\lambda-types and there is a map (A,τ)ctπ(A,τ)H0(T)maps-to𝐴𝜏subscriptct𝜋𝐴𝜏superscriptH0𝑇(A,\tau)\mapsto\mathrm{ct}_{\pi}(A,\tau)\in\mathrm{H}^{0}(T) associating an Azumaya 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}}-algebra with a λ𝜆\lambda-involution to its coarse type, see Subsection 5.2.

The short exact sequence 1R×/S×xxλx1NT11superscript𝑅superscript𝑆maps-to𝑥superscript𝑥𝜆superscript𝑥1𝑁𝑇11\to R^{\times}/S^{\times}\xrightarrow{x\mapsto x^{\lambda}x^{-1}}N\to T\to 1 induces the connecting homomorphism

δ0:H0(T)H1(R×/S×):superscript𝛿0superscriptH0𝑇superscriptH1superscript𝑅superscript𝑆\delta^{0}:\mathrm{H}^{0}(T)\to\mathrm{H}^{1}({R^{\times}}/{S^{\times}})

and the short exact sequence 1S×R×R×/S×11superscript𝑆superscript𝑅superscript𝑅superscript𝑆11\to{S^{\times}}\to{R^{\times}}\to{R^{\times}}/{S^{\times}}\to 1 induces a connecting homomorphism

δ1:H1(R×/S×)H2(S×).:superscript𝛿1superscriptH1superscript𝑅superscript𝑆superscriptH2superscript𝑆\delta^{1}:\mathrm{H}^{1}({R^{\times}}/{S^{\times}})\to\mathrm{H}^{2}({S^{\times}}).
Notation 6.3.1.

We denote the composite morphism δ1δ0superscript𝛿1superscript𝛿0\delta^{1}\circ\delta^{0} by ΦΦ\Phi,

Φ:cTyp(λ)=H0(T)H2(S×).:ΦcTyp𝜆superscriptH0𝑇superscriptH2superscript𝑆\Phi:{\mathrm{cTyp}({\lambda})}=\mathrm{H}^{0}(T)\to\mathrm{H}^{2}({S^{\times}}).
Proposition 6.3.2.

The map ΦΦ\Phi is the 00-map in the following cases:

  1. (i)

    When π:𝐗𝐘:𝜋𝐗𝐘\pi:{\mathbf{X}}\to{\mathbf{Y}} is a trivial quotient (Example 4.3.5), i.e. R=S𝑅𝑆R=S.

  2. (ii)

    When π𝜋\pi is everywhere ramified (Definition 4.5.1), 2S×2superscript𝑆2\in{S^{\times}} and S×superscript𝑆{S^{\times}} has square roots locally.

  3. (iii)

    When π𝜋\pi is unramified (Definition 4.5.1), i.e. R𝑅R is a quadratic étale S𝑆S-algebra.

  4. (iv)

    When π:XY:𝜋𝑋𝑌\pi:X\to Y is a good C2subscript𝐶2C_{2}-quotient of schemes, Y𝑌Y is noetherian and regular, and π𝜋\pi is unramified at the generic points of Y𝑌Y; the corresponding exact quotient is obtained by taking étale ringed topoi as in Example 4.3.3.

Proof.
  1. (i)

    In this case, R×/S×superscript𝑅superscript𝑆{R^{\times}}/{S^{\times}} is trivial. Since ΦΦ\Phi factors through H1(R×/S×)=0superscriptH1superscript𝑅superscript𝑆0\mathrm{H}^{1}({R^{\times}}/{S^{\times}})=0, the result follows.

  2. (ii)

    We claim that squaring induces an automorphism of R×/S×superscript𝑅superscript𝑆{R^{\times}}/{S^{\times}}, and hence of the group H1(R×/S×)superscriptH1superscript𝑅superscript𝑆\mathrm{H}^{1}({R^{\times}}/{S^{\times}}). Since H0(T)superscriptH0𝑇\mathrm{H}^{0}(T) is a 222-torsion group (Proposition 5.2.3), this forces δ0:H0(T)H1(R×/S×):superscript𝛿0superscriptH0𝑇superscriptH1superscript𝑅superscript𝑆\delta^{0}:\mathrm{H}^{0}(T)\to\mathrm{H}^{1}({R^{\times}}/{S^{\times}}) to vanish, implying ΦΦ\Phi vanishes as well.

    We show the surjectivity of xx2:R×/S×R×/S×:maps-to𝑥superscript𝑥2superscript𝑅superscript𝑆superscript𝑅superscript𝑆x\mapsto x^{2}:{R^{\times}}/{S^{\times}}\to{R^{\times}}/{S^{\times}} by checking that R×superscript𝑅{R^{\times}} has square roots locally. Let U𝑈U be an object of 𝐘𝐘{\mathbf{Y}} and rR×(U)𝑟superscript𝑅𝑈r\in{R^{\times}}(U). Since S×superscript𝑆{S^{\times}} has square roots locally, there a covering VU𝑉𝑈V\to U and sS×(V)𝑠superscript𝑆𝑉s\in{S^{\times}}(V) such that rλr=s2superscript𝑟𝜆𝑟superscript𝑠2r^{\lambda}r=s^{2}. Replacing r𝑟r with rs1𝑟superscript𝑠1rs^{-1} and U𝑈U with V𝑉V, we may assume rλr=1superscript𝑟𝜆𝑟1r^{\lambda}r=1. Now, by Lemma 5.3.6, there is a covering {ViU}i=1,2subscriptsubscript𝑉𝑖𝑈𝑖12\{V_{i}\to U\}_{i=1,2} and βiR×(Vi)subscript𝛽𝑖superscript𝑅subscript𝑉𝑖\beta_{i}\in{R^{\times}}(V_{i}) such that r=β11β1λ𝑟superscriptsubscript𝛽11superscriptsubscript𝛽1𝜆r=\beta_{1}^{-1}\beta_{1}^{\lambda} in R×(V1)superscript𝑅subscript𝑉1{R^{\times}}(V_{1}) and r=β21β2λ𝑟superscriptsubscript𝛽21superscriptsubscript𝛽2𝜆r=-\beta_{2}^{-1}\beta_{2}^{\lambda} in R×(V2)superscript𝑅subscript𝑉2{R^{\times}}(V_{2}). We may refine V2Vsubscript𝑉2𝑉V_{2}\to V to assume that there is aS×(V2)𝑎superscript𝑆subscript𝑉2a\in{S^{\times}}(V_{2}) such that β2λβ2=a2superscriptsubscript𝛽2𝜆subscript𝛽2superscript𝑎2-\beta_{2}^{\lambda}\beta_{2}=a^{2} and get r=a2β22𝑟superscript𝑎2superscriptsubscript𝛽22r=a^{2}\beta_{2}^{-2}. Similarly, we refine V1subscript𝑉1V_{1} to find a square root of r𝑟r in R×(V1)superscript𝑅subscript𝑉1{R^{\times}}(V_{1}) and conclude that r𝑟r has a square root on V1V2square-unionsubscript𝑉1subscript𝑉2V_{1}\sqcup V_{2}.

    Next, let K𝐾K denote the kernel of xx2:R×/S×R×/S×:maps-to𝑥superscript𝑥2superscript𝑅superscript𝑆superscript𝑅superscript𝑆x\mapsto x^{2}:{R^{\times}}/{S^{\times}}\to{R^{\times}}/{S^{\times}}. A section of K𝐾K is represented by some aR×(U)𝑎superscript𝑅𝑈a\in{R^{\times}}(U) such that a2S×(U)superscript𝑎2superscript𝑆𝑈a^{2}\in{S^{\times}}(U), or rather, a2=(aλ)2superscript𝑎2superscriptsuperscript𝑎𝜆2a^{2}=(a^{\lambda})^{2}. Since (aaλ)2+(a+aλ)2=4a2S×(U)superscript𝑎superscript𝑎𝜆2superscript𝑎superscript𝑎𝜆24superscript𝑎2superscript𝑆𝑈(a-a^{\lambda})^{2}+(a+a^{\lambda})^{2}=4a^{2}\in{S^{\times}}(U) and (aaλ)2,(a+aλ)2S(U)superscript𝑎superscript𝑎𝜆2superscript𝑎superscript𝑎𝜆2𝑆𝑈(a-a^{\lambda})^{2},(a+a^{\lambda})^{2}\in S(U), and since S𝑆S is a local ring object, there is a covering {UiU}i=1,2subscriptsubscript𝑈𝑖𝑈𝑖12\{U_{i}\to U\}_{i=1,2} such that aaλR×(U1)𝑎superscript𝑎𝜆superscript𝑅subscript𝑈1a-a^{\lambda}\in{R^{\times}}(U_{1}) and a+aλR×(U2)𝑎superscript𝑎𝜆superscript𝑅subscript𝑈2a+a^{\lambda}\in{R^{\times}}(U_{2}). By virtue of Lemma 3.3.3, RU1subscript𝑅subscript𝑈1R_{U_{1}} is a quadratic étale over SU1subscript𝑆subscript𝑈1S_{U_{1}}, so our assumption that π𝜋\pi is everywhere ramified forces U1=subscript𝑈1U_{1}=\emptyset. Thus, U2Usubscript𝑈2𝑈U_{2}\to U is a covering, implying that a+aλ𝑎superscript𝑎𝜆a+a^{\lambda} is invertible in R(U)𝑅𝑈R(U). Since (a+aλ)(aaλ)=a2(aλ)2=0𝑎superscript𝑎𝜆𝑎superscript𝑎𝜆superscript𝑎2superscriptsuperscript𝑎𝜆20(a+a^{\lambda})(a-a^{\lambda})=a^{2}-(a^{\lambda})^{2}=0, we must have aaλ=0𝑎superscript𝑎𝜆0a-a^{\lambda}=0, so aS×(U)𝑎superscript𝑆𝑈a\in{S^{\times}}(U). It follows that a𝑎a represents the 111-section in R×/S×superscript𝑅superscript𝑆{R^{\times}}/{S^{\times}}, and thus K=0𝐾0K=0.

  3. (iii)

    In this case, a version of Hilbert’s Theorem 90 applies in the form of Proposition 5.2.2, and H0(𝐘,T)=0superscriptH0𝐘𝑇0\mathrm{H}^{0}(\mathbf{Y},T)=0. A fortiori, ΦΦ\Phi is 00.

  4. (iv)

    We may assume that Y𝑌Y is connected and therefore integral, otherwise we may work component by component.

    Let ξ:SpecKY:𝜉Spec𝐾𝑌\xi:\operatorname{Spec}K\to Y denote the generic point of Y𝑌Y. Since ξ𝜉\xi is flat, πξ:XξSpecK:subscript𝜋𝜉subscript𝑋𝜉Spec𝐾\pi_{\xi}:X_{\xi}\to\operatorname{Spec}K is a good C2subscript𝐶2C_{2}-quotient relative to the action induced by λ𝜆\lambda; denote the sheaves corresponding to S,R,N,T𝑆𝑅𝑁𝑇S,R,N,T by S,R,N,Tsuperscript𝑆superscript𝑅superscript𝑁superscript𝑇S^{\prime},R^{\prime},N^{\prime},T^{\prime}. We now apply Lemma 5.4.3 to the square

    Xξsubscript𝑋𝜉\textstyle{X_{\xi}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}ξπsubscript𝜉𝜋\scriptstyle{\xi_{\pi}}πξsubscript𝜋𝜉\scriptstyle{\pi_{\xi}}X𝑋\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}π𝜋\scriptstyle{\pi}SpecKSpec𝐾\textstyle{\operatorname{Spec}K\ignorespaces\ignorespaces\ignorespaces\ignorespaces}ξ𝜉\scriptstyle{\xi}Y𝑌\textstyle{Y}

    which gives rise to maps ξNNsuperscript𝜉𝑁superscript𝑁\xi^{*}N\to N^{\prime}, ξTTsuperscript𝜉𝑇superscript𝑇\xi^{*}T\to T^{\prime}, adjoint to the maps in the lemma. The exactness of ξsuperscript𝜉\xi^{*} together with the natural homomorphism H(Y,)H(K,ξ())superscriptH𝑌superscriptH𝐾superscript𝜉\mathrm{H}^{*}(Y,-)\to\mathrm{H}^{*}(K,\xi^{*}(-)) now give rise to a commutative diagram

    Hét0(Y,T)superscriptsubscriptHét0𝑌𝑇\textstyle{\mathrm{H}_{\text{\'{e}t}}^{0}(Y,T)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}ΦΦ\scriptstyle{\Phi}Hét2(Y,𝒪Y×)subscriptsuperscriptH2ét𝑌superscriptsubscript𝒪𝑌\textstyle{\mathrm{H}^{2}_{\text{\'{e}t}}(Y,{{\mathcal{O}}_{Y}^{\times}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Hét0(K,T)subscriptsuperscriptH0ét𝐾superscript𝑇\textstyle{\mathrm{H}^{0}_{\text{\'{e}t}}(K,T^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}ΦξsubscriptΦ𝜉\scriptstyle{\Phi_{\xi}}Hét2(K,𝒪SpecK×)subscriptsuperscriptH2ét𝐾superscriptsubscript𝒪Spec𝐾\textstyle{\mathrm{H}^{2}_{\text{\'{e}t}}(K,{{\mathcal{O}}_{\operatorname{Spec}K}^{\times}})}

    By [grothendieck_groupe_1968, Cor. 1.8], the right vertical morphism is injective (here we need Y𝑌Y to be regular), and by (iii), Φξ=0subscriptΦ𝜉0\Phi_{\xi}=0. Therefore, Φ=0Φ0\Phi=0. ∎

We are now ready to state our generalization of Saltman’s theorem. Whereas Saltman’s original proof [saltman_azumaya_1978] and the later proof by Knus, Parimala and Srinivas [knus_azumaya_1990] make use of the corestriction of an Azumaya algebra, we cannot employ this construction, as demonstrated in Remark 6.2.4. Rather, our proof is purely cohomological, phrased in the language set in Subsections 2.3 and 2.4. We remind the reader of our standing assumption from Remark 4.3.13 that the degrees of all Azumaya 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}}-algebras considered are fixed under ΛΛ\Lambda, which is automatic when 𝐗𝐗{\mathbf{X}} is connected.

Theorem 6.3.3.

Let 𝐗𝐗{\mathbf{X}} be a locally ringed topos with involution λ𝜆\lambda, let π:𝐗𝐘:𝜋𝐗𝐘\pi:{\mathbf{X}}\to{\mathbf{Y}} be an exact quotient relative to λ𝜆\lambda, and consider the map transfλ:Br(𝐗,𝒪𝐗)H2(𝐘,𝒪𝐘×):subscripttransf𝜆Br𝐗subscript𝒪𝐗superscriptH2𝐘superscriptsubscript𝒪𝐘{\operatorname{transf}}_{\lambda}:\operatorname{Br}({\mathbf{X}},{\mathcal{O}}_{\mathbf{X}})\to\mathrm{H}^{2}({\mathbf{Y}},{{\mathcal{O}}_{\mathbf{Y}}^{\times}}) of Definition 6.2.1 and the map Φ:cTyp(λ)=H0(T)H2(𝐘,𝒪𝐘×):ΦcTyp𝜆superscriptH0𝑇superscriptH2𝐘superscriptsubscript𝒪𝐘\Phi:{\mathrm{cTyp}({\lambda})}=\mathrm{H}^{0}(T)\to\mathrm{H}^{2}({\mathbf{Y}},{{\mathcal{O}}_{\mathbf{Y}}^{\times}}) of Notation 6.3.1. Let A𝐴A be an Azumaya 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}}-algebra of degree n𝑛n, and let tH0(T)𝑡superscriptH0𝑇t\in\mathrm{H}^{0}(T). Then there exists A[A]superscript𝐴delimited-[]𝐴A^{\prime}\in[A] admitting a λ𝜆\lambda-involution of coarse type t𝑡t if and only if transf([A])=Φ(t)transfdelimited-[]𝐴Φ𝑡\operatorname{transf}([A])=\Phi(t) in H2(𝐘,𝒪𝐘×)superscriptH2𝐘superscriptsubscript𝒪𝐘\mathrm{H}^{2}({\mathbf{Y}},{{\mathcal{O}}_{\mathbf{Y}}^{\times}}). The algebra Asuperscript𝐴A^{\prime} can be chosen such that degA=2ndegsuperscript𝐴2𝑛\operatorname{deg}A^{\prime}=2n.

We recover Saltman’s original theorem [saltman_azumaya_1978, Th. 3.1] and the improvement of Knus, Parimala and Sinivas [knus_azumaya_1990, Thms. 4.1, 4.2] from Theorem 6.3.3 by taking π:𝐗𝐘:𝜋𝐗𝐘\pi:{\mathbf{X}}\to{\mathbf{Y}} to be the exact quotient associated to a good C2subscript𝐶2C_{2}-quotient of schemes π:XY:𝜋𝑋𝑌\pi:X\to Y such that π𝜋\pi is an isomorphism or quadratic étale, see Examples 4.3.3. In this case, Φ=0Φ0\Phi=0 by Proposition 6.3.2, and the transfer map coincides with multiplication by 222 when π=id𝜋id\pi=\mathrm{id}, or with the corestriction map when π𝜋\pi is quadratic étale, as demonstrated in Examples 6.2.2 and 6.2.3.

The relation between the type and the coarse type of an involution, as well as the question of when two involutions of the same type are locally isomorphic, had been studied extensively in Subsections 5.2 and 5.4.

Proof.

Thanks to Theorems 4.3.6 and 4.3.11, we may replace A𝐴A with πAsubscript𝜋𝐴\pi_{*}A and work with R𝑅R-algebras, rather than 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}}-algebras. We abuse the notation and denote the map H2(R×)H2(S×)superscriptH2superscript𝑅superscriptH2superscript𝑆\mathrm{H}^{2}({R^{\times}})\to\mathrm{H}^{2}({S^{\times}}) induced by xxλx:R×S×:maps-to𝑥superscript𝑥𝜆𝑥superscript𝑅superscript𝑆x\mapsto x^{\lambda}x:{R^{\times}}\to{S^{\times}} as transfλsubscripttransf𝜆\operatorname{transf}_{\lambda}.

Suppose first that there exists [A]Adelimited-[]superscript𝐴𝐴[A^{\prime}]\in A admitting a λ𝜆\lambda-involution τ𝜏\tau of coarse type t𝑡t. We may replace A𝐴A with Asuperscript𝐴A^{\prime}. We now invoke all the notation of Construction 5.2.5 and the proof of Lemma 5.2.6 through which t𝑡t is constructed from (A,τ)𝐴𝜏(A,\tau). Specifically:

  • U𝐘𝑈subscript𝐘U\to*_{\mathbf{Y}} is a covering such that there exists an isomorphism of RUsubscript𝑅𝑈R_{U}-algebras ψ:AUMn×n(RU):𝜓subscript𝐴𝑈subscriptM𝑛𝑛subscript𝑅𝑈\psi:A_{U}\to\mathrm{M}_{n\times n}(R_{U}),

  • σ:=ψτUψ1assign𝜎𝜓subscript𝜏𝑈superscript𝜓1\sigma:=\psi\circ\tau_{U}\circ\psi^{-1} is an involution of Mn×n(RU)subscriptM𝑛𝑛subscript𝑅𝑈\mathrm{M}_{n\times n}(R_{U}),

  • g:=λtrσassign𝑔𝜆tr𝜎g:=\lambda\text{\rm tr}\circ\sigma is an element of PGLn(R)(U)subscriptPGL𝑛𝑅𝑈\operatorname{PGL}_{n}(R)(U),

  • hGLn(R)(U)subscriptGL𝑛𝑅𝑈h\in\operatorname{GL}_{n}(R)(U) is a lift of g𝑔g (refine U𝑈U if necessary),

  • ε:=hλtrhassign𝜀superscript𝜆tr\varepsilon:=h^{-\lambda\text{\rm tr}}h is an element of N(U)𝑁𝑈N(U), embedded diagonally in GLn(R)(U)subscriptGL𝑛𝑅𝑈\operatorname{GL}_{n}(R)(U),

  • Usubscript𝑈U_{\bullet} is the Čech hypercovering corresponding to U𝑈U\to*, see Example 2.3.1,

  • a=ψ1ψ01PGLn(R)(U1)𝑎subscript𝜓1superscriptsubscript𝜓01subscriptPGL𝑛𝑅subscript𝑈1a=\psi_{1}\circ\psi_{0}^{-1}\in\operatorname{PGL}_{n}(R)(U_{1}), where ψisubscript𝜓𝑖\psi_{i} is the pullback of ψ𝜓\psi along di:U1U0:subscript𝑑𝑖subscript𝑈1subscript𝑈0d_{i}:U_{1}\to U_{0},

  • b𝑏b is a lift of a𝑎a to GLn(R)(V)subscriptGL𝑛𝑅𝑉\operatorname{GL}_{n}(R)(V), where VU1𝑉subscript𝑈1V\to U_{1} is some covering,

  • β:=bλtrd0hb1d1h1assign𝛽superscript𝑏𝜆trsuperscriptsubscript𝑑0superscript𝑏1superscriptsubscript𝑑1superscript1\beta:=b^{-\lambda\text{\rm tr}}\cdot d_{0}^{*}h\cdot b^{-1}\cdot d_{1}^{*}h^{-1} is an element R×(V)superscript𝑅𝑉{R^{\times}}(V), embedded diagonally in GLn(R)(V)subscriptGL𝑛𝑅𝑉\operatorname{GL}_{n}(R)(V),

  • t=ct(τ)𝑡ct𝜏t=\mathrm{ct}(\tau) is the image of ε𝜀\varepsilon in T(U)𝑇𝑈T(U); it descends to a global section of T𝑇T since d1εd0ε1=β1βλsuperscriptsubscript𝑑1𝜀superscriptsubscript𝑑0superscript𝜀1superscript𝛽1superscript𝛽𝜆d_{1}^{*}\varepsilon\cdot d_{0}^{*}\varepsilon^{-1}=\beta^{-1}\beta^{\lambda}.

By Lemma 2.3.2, there is a hypercovering morphism VUsubscript𝑉subscript𝑈V_{\bullet}\to U_{\bullet} such that V1U1subscript𝑉1subscript𝑈1V_{1}\to U_{1} factors through VU1𝑉subscript𝑈1V\to U_{1}. We replace V𝑉V with V1subscript𝑉1V_{1}.

Recall from Theorem 4.3.11 that A𝐴A corresponds to a PGLn(R)subscriptPGL𝑛𝑅\operatorname{PGL}_{n}(R)-torsor, which in turn corresponds to a cohomology class in H1(PGLn(R))superscriptH1subscriptPGL𝑛𝑅\mathrm{H}^{1}(\operatorname{PGL}_{n}(R)). We claim that a𝑎a is a 111-cocycle in Z1(U,PGLn(R))superscript𝑍1subscript𝑈subscriptPGL𝑛𝑅Z^{1}(U_{\bullet},\operatorname{PGL}_{n}(R)) which represents this cohomology class. Indeed, the PGLn(R)subscriptPGL𝑛𝑅\operatorname{PGL}_{n}(R)-torsor corresponding to a𝑎a is P:=𝒜utR(Mn×n(R),A)assign𝑃𝒜𝑢subscript𝑡𝑅subscriptM𝑛𝑛𝑅𝐴P:=\mathcal{A}ut_{R}(\mathrm{M}_{n\times n}(R),A), and ψ1P(U)=P(U0)superscript𝜓1𝑃𝑈𝑃subscript𝑈0\psi^{-1}\in P(U)=P(U_{0}) by construction. By the isomorphism given in the proof of Proposition 2.4.2(i), the cohomology class corresponding to P𝑃P is represented by d1(ψ1)1d0(ψ1)=ψ1ψ01=asuperscriptsubscript𝑑1superscriptsuperscript𝜓11superscriptsubscript𝑑0superscript𝜓1subscript𝜓1superscriptsubscript𝜓01𝑎d_{1}^{*}(\psi^{-1})^{-1}\cdot d_{0}^{*}(\psi^{-1})=\psi_{1}\circ\psi_{0}^{-1}=a.

Consider the short exact sequence 1R×GLn(R)PGLn(R)11superscript𝑅subscriptGL𝑛𝑅subscriptPGL𝑛𝑅11\to{R^{\times}}\to\operatorname{GL}_{n}(R)\to\operatorname{PGL}_{n}(R)\to 1 and its associated 777-term cohomology exact sequence, see Proposition 2.4.2(iii). It follows from the definition of δ2:H1(PGLn(R))H2(R×):superscript𝛿2superscriptH1subscriptPGL𝑛𝑅superscriptH2superscript𝑅\delta^{2}:\mathrm{H}^{1}(\operatorname{PGL}_{n}(R))\to\mathrm{H}^{2}({R^{\times}}), see the proof of Proposition 2.4.2(iii), that [A]=δ2(a)H2(R×)delimited-[]𝐴superscript𝛿2𝑎superscriptH2superscript𝑅[A]=\delta^{2}(a)\in\mathrm{H}^{2}({R^{\times}}) is represented by

(13) α:=d2bd0bd1b1Z2(V,R×).assign𝛼superscriptsubscript𝑑2𝑏superscriptsubscript𝑑0𝑏superscriptsubscript𝑑1superscript𝑏1superscript𝑍2subscript𝑉superscript𝑅\alpha:=d_{2}^{*}b\cdot d_{0}^{*}b\cdot d_{1}^{*}b^{-1}\in Z^{2}(V_{\bullet},{R^{\times}})\ .

and thus, transf([A])transfdelimited-[]𝐴\operatorname{transf}([A]) is represented by αλαZ2(V,S×)superscript𝛼𝜆𝛼superscript𝑍2subscript𝑉superscript𝑆\alpha^{\lambda}\alpha\in Z^{2}(V_{\bullet},{S^{\times}}).

On the other hand, by the definition of δ0:H0(T)H1(R×/S×):superscript𝛿0superscriptH0𝑇superscriptH1superscript𝑅superscript𝑆\delta^{0}:\mathrm{H}^{0}(T)\to\mathrm{H}^{1}({R^{\times}}/{S^{\times}}), see the beginning of this subsection and the end of 2.3, δ0(t)superscript𝛿0𝑡\delta^{0}(t) is represented by the image of β1R×(V1)superscript𝛽1superscript𝑅subscript𝑉1\beta^{-1}\in{R^{\times}}(V_{1}) in (R×/S×)(V1)superscript𝑅superscript𝑆subscript𝑉1({R^{\times}}/{S^{\times}})(V_{1}), since d0εd1ε1=(β1)λβsuperscriptsubscript𝑑0𝜀superscriptsubscript𝑑1superscript𝜀1superscriptsuperscript𝛽1𝜆𝛽d_{0}^{*}\varepsilon\cdot d_{1}^{*}\varepsilon^{-1}=(\beta^{-1})^{\lambda}\beta. Likewise, by the definition of δ1:H1(R×/S×)H2(S×):superscript𝛿1superscriptH1superscript𝑅superscript𝑆superscriptH2superscript𝑆\delta^{1}:\mathrm{H}^{1}({R^{\times}}/{S^{\times}})\to\mathrm{H}^{2}({S^{\times}}), the class Φ(t)=δ1δ0(t)Φ𝑡superscript𝛿1superscript𝛿0𝑡\Phi(t)=\delta^{1}\delta^{0}(t) is represented by d0β1d1βd2β1Z2(V,S×)superscriptsubscript𝑑0superscript𝛽1superscriptsubscript𝑑1𝛽superscriptsubscript𝑑2superscript𝛽1superscript𝑍2subscript𝑉superscript𝑆d_{0}^{*}\beta^{-1}\cdot d_{1}^{*}\beta\cdot d_{2}^{*}\beta^{-1}\in Z^{2}(V_{\bullet},{S^{\times}}).

In order to show that transf([A])=Φ(t)transfdelimited-[]𝐴Φ𝑡\operatorname{transf}([A])=\Phi(t), we check that αλα=d0β1d1βd2β1superscript𝛼𝜆𝛼superscriptsubscript𝑑0superscript𝛽1superscriptsubscript𝑑1𝛽superscriptsubscript𝑑2superscript𝛽1\alpha^{\lambda}\alpha=d_{0}^{*}\beta^{-1}\cdot d_{1}^{*}\beta\cdot d_{2}^{*}\beta^{-1} in S×(V2)superscript𝑆subscript𝑉2{S^{\times}}(V_{2}). For the computation, we shall make use of d0d0=d1d0superscriptsubscript𝑑0superscriptsubscript𝑑0superscriptsubscript𝑑1superscriptsubscript𝑑0d_{0}^{*}d_{0}^{*}=d_{1}^{*}d_{0}^{*}, d0d1=d2d0superscriptsubscript𝑑0superscriptsubscript𝑑1superscriptsubscript𝑑2superscriptsubscript𝑑0d_{0}^{*}d_{1}^{*}=d_{2}^{*}d_{0}^{*}, d1d1=d2d1superscriptsubscript𝑑1superscriptsubscript𝑑1superscriptsubscript𝑑2superscriptsubscript𝑑1d_{1}^{*}d_{1}^{*}=d_{2}^{*}d_{1}^{*} and the fact that if xyz𝑥𝑦𝑧xyz is central in a group G𝐺G, then xyz=zxy=yzx𝑥𝑦𝑧𝑧𝑥𝑦𝑦𝑧𝑥xyz=zxy=yzx.

d0β1d1βd2β1superscriptsubscript𝑑0superscript𝛽1superscriptsubscript𝑑1𝛽superscriptsubscript𝑑2superscript𝛽1\displaystyle d_{0}^{*}\beta^{-1}\cdot d_{1}^{*}\beta\cdot d_{2}^{*}\beta^{-1} =d0β1(d1bλtrd1d0hd1b1d1d1h1)absentsuperscriptsubscript𝑑0superscript𝛽1superscriptsubscript𝑑1superscript𝑏𝜆trsuperscriptsubscript𝑑1superscriptsubscript𝑑0superscriptsubscript𝑑1superscript𝑏1superscriptsubscript𝑑1superscriptsubscript𝑑1superscript1\displaystyle=d_{0}^{*}\beta^{-1}(d_{1}^{*}b^{-\lambda\text{\rm tr}}\cdot d_{1}^{*}d_{0}^{*}h\cdot d_{1}^{*}b^{-1}\cdot d_{1}^{*}d_{1}^{*}h^{-1})
(d2d1hd2bd2d0h1d2bλtr)absentsuperscriptsubscript𝑑2superscriptsubscript𝑑1superscriptsubscript𝑑2𝑏superscriptsubscript𝑑2superscriptsubscript𝑑0superscript1superscriptsubscript𝑑2superscript𝑏𝜆tr\displaystyle\phantom{=}\quad\cdot(d_{2}^{*}d_{1}^{*}h\cdot d_{2}^{*}b\cdot d_{2}^{*}d_{0}^{*}h^{-1}\cdot d_{2}^{*}b^{\lambda\text{\rm tr}})
=d1bλtrd1d0hd1b1d2bd2d0h1(d0β1)d2bλtrabsentsuperscriptsubscript𝑑1superscript𝑏𝜆trsuperscriptsubscript𝑑1superscriptsubscript𝑑0superscriptsubscript𝑑1superscript𝑏1superscriptsubscript𝑑2𝑏superscriptsubscript𝑑2superscriptsubscript𝑑0superscript1superscriptsubscript𝑑0superscript𝛽1superscriptsubscript𝑑2superscript𝑏𝜆tr\displaystyle=d_{1}^{*}b^{-\lambda\text{\rm tr}}\cdot d_{1}^{*}d_{0}^{*}h\cdot d_{1}^{*}b^{-1}\cdot d_{2}^{*}b\cdot d_{2}^{*}d_{0}^{*}h^{-1}\cdot(d_{0}^{*}\beta^{-1})\cdot d_{2}^{*}b^{\lambda\text{\rm tr}}
=d1bλtrd1d0hd1b1d2bd2d0h1absentsuperscriptsubscript𝑑1superscript𝑏𝜆trsuperscriptsubscript𝑑1superscriptsubscript𝑑0superscriptsubscript𝑑1superscript𝑏1superscriptsubscript𝑑2𝑏superscriptsubscript𝑑2superscriptsubscript𝑑0superscript1\displaystyle=d_{1}^{*}b^{-\lambda\text{\rm tr}}\cdot d_{1}^{*}d_{0}^{*}h\cdot d_{1}^{*}b^{-1}\cdot d_{2}^{*}b\cdot d_{2}^{*}d_{0}^{*}h^{-1}
(d0d1hd0bd0d0h1d0bλtr)d2bλtrabsentsuperscriptsubscript𝑑0superscriptsubscript𝑑1superscriptsubscript𝑑0𝑏superscriptsubscript𝑑0superscriptsubscript𝑑0superscript1superscriptsubscript𝑑0superscript𝑏𝜆trsuperscriptsubscript𝑑2superscript𝑏𝜆tr\displaystyle\phantom{=}\quad\cdot(d_{0}^{*}d_{1}^{*}h\cdot d_{0}^{*}b\cdot d_{0}^{*}d_{0}^{*}h^{-1}\cdot d_{0}^{*}b^{\lambda\text{\rm tr}})\cdot d_{2}^{*}b^{\lambda\text{\rm tr}}
=d1bλtrd1d0h(d1b1d2bd0b)d0d0h1d0bλtrd2bλtrabsentsuperscriptsubscript𝑑1superscript𝑏𝜆trsuperscriptsubscript𝑑1superscriptsubscript𝑑0superscriptsubscript𝑑1superscript𝑏1superscriptsubscript𝑑2𝑏superscriptsubscript𝑑0𝑏superscriptsubscript𝑑0superscriptsubscript𝑑0superscript1superscriptsubscript𝑑0superscript𝑏𝜆trsuperscriptsubscript𝑑2superscript𝑏𝜆tr\displaystyle=d_{1}^{*}b^{-\lambda\text{\rm tr}}\cdot d_{1}^{*}d_{0}^{*}h\cdot(d_{1}^{*}b^{-1}\cdot d_{2}^{*}b\cdot d_{0}^{*}b)\cdot d_{0}^{*}d_{0}^{*}h^{-1}\cdot d_{0}^{*}b^{\lambda\text{\rm tr}}\cdot d_{2}^{*}b^{\lambda\text{\rm tr}}
=d1bλtrd1d0hαd0d0h1d0bλtrd2bλtrabsentsuperscriptsubscript𝑑1superscript𝑏𝜆trsuperscriptsubscript𝑑1superscriptsubscript𝑑0𝛼superscriptsubscript𝑑0superscriptsubscript𝑑0superscript1superscriptsubscript𝑑0superscript𝑏𝜆trsuperscriptsubscript𝑑2superscript𝑏𝜆tr\displaystyle=d_{1}^{*}b^{-\lambda\text{\rm tr}}\cdot d_{1}^{*}d_{0}^{*}h\cdot\alpha\cdot d_{0}^{*}d_{0}^{*}h^{-1}\cdot d_{0}^{*}b^{\lambda\text{\rm tr}}\cdot d_{2}^{*}b^{\lambda\text{\rm tr}}
=(d2bd0bd1b1)λtrαabsentsuperscriptsuperscriptsubscript𝑑2𝑏superscriptsubscript𝑑0𝑏superscriptsubscript𝑑1superscript𝑏1𝜆tr𝛼\displaystyle=(d_{2}^{*}b\cdot d_{0}^{*}b\cdot d_{1}^{*}b^{-1})^{\lambda\text{\rm tr}}\cdot\alpha
=αλαabsentsuperscript𝛼𝜆𝛼\displaystyle=\alpha^{\lambda}\alpha

This completes the proof of the “only if” statement.

Suppose now that transf([A])=Φ(t)transfdelimited-[]𝐴Φ𝑡\operatorname{transf}([A])=\Phi(t). Define U𝑈U\to*, Usubscript𝑈U_{\bullet}, a𝑎a, b𝑏b, Vsubscript𝑉V_{\bullet} and α𝛼\alpha as before. Using Lemma 2.3.2 twice, we can refine Vsubscript𝑉V_{\bullet} to assume that t𝑡t lifts to some εN(V0)𝜀𝑁subscript𝑉0\varepsilon\in N(V_{0}) and there is βR×(V1)𝛽superscript𝑅subscript𝑉1\beta\in{R^{\times}}(V_{1}) such that

(14) d0εd1ε1=(β1)λβsuperscriptsubscript𝑑0𝜀superscriptsubscript𝑑1superscript𝜀1superscriptsuperscript𝛽1𝜆𝛽d_{0}^{*}\varepsilon\cdot d_{1}^{*}\varepsilon^{-1}=(\beta^{-1})^{\lambda}\beta

in N(V1)𝑁subscript𝑉1N(V_{1}). As explained above, transf([A])transfdelimited-[]𝐴\operatorname{transf}([A]) is represented by αλαZ2(V,S×)superscript𝛼𝜆𝛼superscript𝑍2subscript𝑉superscript𝑆\alpha^{\lambda}\alpha\in Z^{2}(V_{\bullet},{S^{\times}}) and Φ(t)Φ𝑡\Phi(t) is represented by d0β1d1βd2β1superscriptsubscript𝑑0superscript𝛽1superscriptsubscript𝑑1𝛽superscriptsubscript𝑑2superscript𝛽1d_{0}^{*}\beta^{-1}\cdot d_{1}^{*}\beta\cdot d_{2}^{*}\beta^{-1}. The assumption Φ(t)=transf([A])Φ𝑡transfdelimited-[]𝐴\Phi(t)=\operatorname{transf}([A]) therefore means that, after refining Vsubscript𝑉V_{\bullet}, there exists γS×(V1)𝛾superscript𝑆subscript𝑉1\gamma\in{S^{\times}}(V_{1}) such that

d0γd1γ1d2γd0β1d1βd2β1=αλα.superscriptsubscript𝑑0𝛾superscriptsubscript𝑑1superscript𝛾1superscriptsubscript𝑑2𝛾superscriptsubscript𝑑0superscript𝛽1superscriptsubscript𝑑1𝛽superscriptsubscript𝑑2superscript𝛽1superscript𝛼𝜆𝛼d_{0}^{*}\gamma\cdot d_{1}^{*}\gamma^{-1}\cdot d_{2}^{*}\gamma\cdot d_{0}^{*}\beta^{-1}\cdot d_{1}^{*}\beta\cdot d_{2}^{*}\beta^{-1}=\alpha^{\lambda}\alpha\ .

We replace β𝛽\beta with βγ1R×(V1)𝛽superscript𝛾1superscript𝑅subscript𝑉1\beta\gamma^{-1}\in{R^{\times}}(V_{1}), which does not affect (14) and allows us to assume

(15) d0β1d1βd2β1=αλα.superscriptsubscript𝑑0superscript𝛽1superscriptsubscript𝑑1𝛽superscriptsubscript𝑑2superscript𝛽1superscript𝛼𝜆𝛼d_{0}^{*}\beta^{-1}\cdot d_{1}^{*}\beta\cdot d_{2}^{*}\beta^{-1}=\alpha^{\lambda}\alpha\ .

Writing in block form, define the 2n×2n2𝑛2𝑛2n\times 2n matrices

h=[01ε0]GL2n(R)(V0)andb=[b00β1bλtr]GL2n(R)(V1)formulae-sequencematrix01𝜀0subscriptGL2𝑛𝑅subscript𝑉0andsuperscript𝑏matrix𝑏00superscript𝛽1superscript𝑏𝜆trsubscriptGL2𝑛𝑅subscript𝑉1h=\begin{bmatrix}0&1\\ \varepsilon&0\end{bmatrix}\in\operatorname{GL}_{2n}(R)(V_{0})\qquad\text{and}\qquad b^{\prime}=\begin{bmatrix}b&0\\ 0&\beta^{-1}b^{-\lambda\text{\rm tr}}\end{bmatrix}\in\operatorname{GL}_{2n}(R)(V_{1})

and let σ:M2n×2n(RV0)M2n×2n(RV0):𝜎subscriptM2𝑛2𝑛subscript𝑅subscript𝑉0subscriptM2𝑛2𝑛subscript𝑅subscript𝑉0\sigma:\mathrm{M}_{2n\times 2n}(R_{V_{0}})\to\mathrm{M}_{2n\times 2n}(R_{V_{0}}) be the involution given by x(hxh1)λtr=hλtrxhλtrmaps-to𝑥superscript𝑥superscript1𝜆trsuperscript𝜆tr𝑥superscript𝜆trx\mapsto(hxh^{-1})^{\lambda\text{\rm tr}}=h^{-\lambda\text{\rm tr}}xh^{\lambda\text{\rm tr}} on sections. Also, let asuperscript𝑎a^{\prime} be the image of bsuperscript𝑏b^{\prime} in PGL2n(R)(V1)subscriptPGL2𝑛𝑅subscript𝑉1\operatorname{PGL}_{2n}(R)(V_{1}), namely, aPGL2n(R)(V1)superscript𝑎subscriptPGL2𝑛𝑅subscript𝑉1a^{\prime}\in\operatorname{PGL}_{2n}(R)(V_{1}) is the automorphism of M2n×2n(RV1)subscriptM2𝑛2𝑛subscript𝑅subscript𝑉1\mathrm{M}_{2n\times 2n}(R_{V_{1}}) given by xbxb1maps-to𝑥superscript𝑏𝑥superscript𝑏1x\mapsto b^{\prime}xb^{\prime-1} on sections.

We first observe that aZ1(V,PGL2n(R))superscript𝑎superscript𝑍1subscript𝑉subscriptPGL2𝑛𝑅a^{\prime}\in Z^{1}(V_{\bullet},\operatorname{PGL}_{2n}(R)). Indeed, working in GL2n(R)(V2)subscriptGL2𝑛𝑅subscript𝑉2\operatorname{GL}_{2n}(R)(V_{2}) and using (13) and (15), we find that

(16) d2bd0bd1b1=[α00(d0βd2βd1β1)1αλtr]=[α00α]R×(V1).superscriptsubscript𝑑2superscript𝑏superscriptsubscript𝑑0superscript𝑏superscriptsubscript𝑑1superscript𝑏1matrix𝛼00superscriptsuperscriptsubscript𝑑0𝛽superscriptsubscript𝑑2𝛽superscriptsubscript𝑑1superscript𝛽11superscript𝛼𝜆trmatrix𝛼00𝛼superscript𝑅subscript𝑉1d_{2}^{*}b^{\prime}\cdot d_{0}^{*}b^{\prime}\cdot d_{1}^{*}b^{\prime-1}=\begin{bmatrix}\alpha&0\\ 0&(d_{0}^{*}\beta\cdot d_{2}^{*}\beta\cdot d_{1}^{*}\beta^{-1})^{-1}\alpha^{-\lambda\text{\rm tr}}\end{bmatrix}=\begin{bmatrix}\alpha&0\\ 0&\alpha\end{bmatrix}\in{R^{\times}}(V_{1})\ .

Let V~subscript~𝑉\tilde{V}_{\bullet} denote the Čech hypercovering associated to V0subscript𝑉0V_{0}\to*, see Example 2.3.1. By Lemma 2.4.1, aZ1(V,PGL2n(R))superscript𝑎superscript𝑍1subscript𝑉subscriptPGL2𝑛𝑅a^{\prime}\in Z^{1}(V_{\bullet},\operatorname{PGL}_{2n}(R)) descends uniquely to a cocycle a~Z1(V~,PGL2n(R))superscript~𝑎superscript𝑍1subscript~𝑉subscriptPGL2𝑛𝑅\tilde{a}^{\prime}\in Z^{1}(\tilde{V}_{\bullet},\operatorname{PGL}_{2n}(R)). The Čech 111-cocycle a~superscript~𝑎\tilde{a}^{\prime} defines descent data for M2n×2n(RV0)subscriptM2𝑛2𝑛subscript𝑅subscript𝑉0\mathrm{M}_{2n\times 2n}(R_{V_{0}}) along V0subscript𝑉0V_{0}\to*, giving rise to an Azumaya R𝑅R-algebra Asuperscript𝐴A^{\prime} of degree 2n2𝑛2n and an isomorphism ψ:AV0M2n×2n(RV0):𝜓subscriptsuperscript𝐴subscript𝑉0subscriptM2𝑛2𝑛subscript𝑅subscript𝑉0\psi:A^{\prime}_{V_{0}}\to\mathrm{M}_{2n\times 2n}(R_{V_{0}}) such that a~=ψ1ψ01superscript~𝑎subscript𝜓1superscriptsubscript𝜓01\tilde{a}^{\prime}=\psi_{1}\circ\psi_{0}^{-1}, where ψisubscript𝜓𝑖\psi_{i} is the pullback of ψ𝜓\psi along di:V~1V~0=V0:subscript𝑑𝑖subscript~𝑉1subscript~𝑉0subscript𝑉0d_{i}:\tilde{V}_{1}\to\tilde{V}_{0}=V_{0}. Note that by construction, asuperscript𝑎a^{\prime} represents the class in H1(PGL2n(R))superscriptH1subscriptPGL2𝑛𝑅\mathrm{H}^{1}(\operatorname{PGL}_{2n}(R)) corresponding to Asuperscript𝐴A^{\prime}, hence (16) implies that [A]=α=[A]delimited-[]superscript𝐴𝛼delimited-[]𝐴[A^{\prime}]=\alpha=[A].

We now claim that σ𝜎\sigma descends to an involution τ:AA:𝜏superscript𝐴superscript𝐴\tau:A^{\prime}\to A^{\prime}. Letting σisubscript𝜎𝑖\sigma_{i} denote the pullback of σ𝜎\sigma along di:V1V0:subscript𝑑𝑖subscript𝑉1subscript𝑉0d_{i}:V_{1}\to V_{0}, and noting that (d0,d1):V1V0×V0:subscript𝑑0subscript𝑑1subscript𝑉1subscript𝑉0subscript𝑉0(d_{0},d_{1}):V_{1}\to V_{0}\times V_{0} is a covering, see Subsection 2.3, this amounts to showing that σ1a=aσ0subscript𝜎1superscript𝑎superscript𝑎subscript𝜎0\sigma_{1}a^{\prime}=a^{\prime}\sigma_{0}. To see this, we first note that (14) and ελε=1superscript𝜀𝜆𝜀1\varepsilon^{\lambda}\varepsilon=1 imply that

bλtrd0hb1d1h1=[β00βλd0εd1ε1]=[β00β],superscript𝑏𝜆trsuperscriptsubscript𝑑0superscript𝑏1superscriptsubscript𝑑1superscript1matrix𝛽00superscript𝛽𝜆superscriptsubscript𝑑0𝜀superscriptsubscript𝑑1superscript𝜀1matrix𝛽00𝛽b^{\prime-\lambda\text{\rm tr}}\cdot d_{0}^{*}h\cdot b^{\prime-1}\cdot d_{1}^{*}h^{-1}=\begin{bmatrix}\beta&0\\ 0&\beta^{\lambda}\cdot d_{0}^{*}\varepsilon\cdot d_{1}^{*}\varepsilon^{-1}\end{bmatrix}=\begin{bmatrix}\beta&0\\ 0&\beta\end{bmatrix},

or equivalently,

b1d1h1=βd0h1bλtr.superscript𝑏1superscriptsubscript𝑑1superscript1𝛽superscriptsubscript𝑑0superscript1superscript𝑏𝜆trb^{\prime-1}\cdot d_{1}^{*}h^{-1}=\beta\cdot d_{0}^{*}h^{-1}\cdot b^{\prime\lambda\text{\rm tr}}.

Using this, for any section x𝑥x of M2n×2n(RV1)subscriptM2𝑛2𝑛subscript𝑅subscript𝑉1\mathrm{M}_{2n\times 2n}(R_{V_{1}}), we have

σ1(a(x))subscript𝜎1superscript𝑎𝑥\displaystyle\sigma_{1}(a^{\prime}(x)) =d1hλtr(bxb1)λtrd1hλtr=(b1d1h1)λtrx(b1d1h1)λtrabsentsuperscriptsubscript𝑑1superscript𝜆trsuperscriptsuperscript𝑏𝑥superscript𝑏1𝜆trsuperscriptsubscript𝑑1superscript𝜆trsuperscriptsuperscript𝑏1superscriptsubscript𝑑1superscript1𝜆tr𝑥superscriptsuperscript𝑏1superscriptsubscript𝑑1superscript1𝜆tr\displaystyle=d_{1}^{*}h^{-\lambda\text{\rm tr}}(b^{\prime}xb^{\prime-1})^{\lambda\text{\rm tr}}d_{1}^{*}h^{\lambda\text{\rm tr}}=(b^{\prime-1}d_{1}^{*}h^{-1})^{\lambda\text{\rm tr}}x(b^{\prime-1}d_{1}^{*}h^{-1})^{-\lambda\text{\rm tr}}
=(βd0h1bλtr)λtrx(βd0h1bλtr)λtr=a(σ0(x)),absentsuperscript𝛽superscriptsubscript𝑑0superscript1superscript𝑏𝜆tr𝜆tr𝑥superscript𝛽superscriptsubscript𝑑0superscript1superscript𝑏𝜆tr𝜆trsuperscript𝑎subscript𝜎0𝑥\displaystyle=(\beta\cdot d_{0}^{*}h^{-1}\cdot b^{\prime\lambda\text{\rm tr}})^{\lambda\text{\rm tr}}x(\beta\cdot d_{0}^{*}h^{-1}\cdot b^{\prime\lambda\text{\rm tr}})^{-\lambda\text{\rm tr}}=a^{\prime}(\sigma_{0}(x))\ ,

which is what we want.

We finish by checking that ct(τ)=tct𝜏𝑡\mathrm{ct}(\tau)=t. To see this, apply Construction 5.2.5 to (A,τ)superscript𝐴𝜏(A^{\prime},\tau) using U:=V0assign𝑈subscript𝑉0U:=V_{0}, ψ𝜓\psi, σ𝜎\sigma and hh defined above and note that hλtrh=[ε00ε]superscript𝜆trdelimited-[]𝜀00𝜀h^{-\lambda\text{\rm tr}}h=[\begin{smallmatrix}\varepsilon&0\\ 0&\varepsilon\end{smallmatrix}]. ∎

We now specialize Theorem 6.3.3 to Azumaya algebras over schemes and over topological spaces.

It is worth recalling at this point that in the situation of a good C2subscript𝐶2C_{2}-quotients of schemes π:XY:𝜋𝑋𝑌\pi:X\to Y such that 222 is invertible on Y𝑌Y (Example 4.3.3), or a C2subscript𝐶2C_{2}-quotient of Hausdorff topological spaces π:XY:𝜋𝑋𝑌\pi:X\to Y (Examples 4.3.4), the sheaf T𝑇T is isomorphic to iμ2,Wsubscript𝑖subscript𝜇2𝑊i_{*}\mu_{2,W}, where i:WY:𝑖𝑊𝑌i:W\to Y is the embedding of the branch locus of π𝜋\pi in Y𝑌Y. Under this isomorphism, the coarse type of an involution τ:AA:𝜏𝐴𝐴\tau:A\to A is the unique global section fH0(W,μ2,W)=𝒞(W,{±1})𝑓superscriptH0𝑊subscript𝜇2𝑊𝒞𝑊plus-or-minus1f\in\mathrm{H}^{0}(W,\mu_{2,W})={\mathcal{C}}(W,\{\pm 1\}) such that f(w)=1𝑓𝑤1f(w)=1 if τk(π1(w)):Ak(π1(w))Ak(π1(w)):subscript𝜏𝑘superscript𝜋1𝑤subscript𝐴𝑘superscript𝜋1𝑤subscript𝐴𝑘superscript𝜋1𝑤\tau_{k(\pi^{-1}(w))}:A_{k(\pi^{-1}(w))}\to A_{k(\pi^{-1}(w))} is orthogonal, and f(w)=1𝑓𝑤1f(w)=-1 if τk(π1(w)):Ak(π1(w))Ak(π1(w)):subscript𝜏𝑘superscript𝜋1𝑤subscript𝐴𝑘superscript𝜋1𝑤subscript𝐴𝑘superscript𝜋1𝑤\tau_{k(\pi^{-1}(w))}:A_{k(\pi^{-1}(w))}\to A_{k(\pi^{-1}(w))} is symplectic, for all wW𝑤𝑊w\in W; see Subsection 5.4. Furthermore, in these situations, two λ𝜆\lambda-involutions of the same coarse type have the same type, and they are locally isomorphic if the degrees of their underlying Azumaya algebras agree; this follows from Theorem 5.2.13 and Corollary 5.2.14.

Corollary 6.3.4.

Let X𝑋X be a scheme, let λ:XX:𝜆𝑋𝑋\lambda:X\to X be an involution and let π:XY:𝜋𝑋𝑌\pi:X\to Y be a good quotient relative to C2:={1,λ}assignsubscript𝐶21𝜆C_{2}:=\{1,\lambda\}. Let A𝐴A be an Azumaya 𝒪Xsubscript𝒪𝑋{\mathcal{O}}_{X}-algebra and let tcTyp(λ)𝑡cTyp𝜆t\in{\mathrm{cTyp}({\lambda})} be a coarse type. Then there exists A[A]superscript𝐴delimited-[]𝐴A^{\prime}\in[A] admitting a λ𝜆\lambda-involution of coarse type t𝑡t if and only if Φ(t)=transfλ([A])Φ𝑡subscripttransf𝜆delimited-[]𝐴\Phi(t)=\operatorname{transf}_{\lambda}([A]) in Hét2(Y,𝒪Y×)subscriptsuperscriptH2ét𝑌superscriptsubscript𝒪𝑌\mathrm{H}^{2}_{\text{\'{e}t}}(Y,{{\mathcal{O}}_{Y}^{\times}}). The algebra Asuperscript𝐴A^{\prime} can be chosen such that degA=2degAdegsuperscript𝐴2deg𝐴\operatorname{deg}A^{\prime}=2\operatorname{deg}A.

Proof.

This is a special case of Theorem 6.3.3, see Example 4.3.3 and Theorem 4.4.4. ∎

Corollary 6.3.5.

In the situation of Corollary 6.3.4, suppose that

  1. (1)

    λ=id𝜆id\lambda=\mathrm{id}, or

  2. (2)

    π:XY:𝜋𝑋𝑌\pi:X\to Y is quadratic étale, or

  3. (3)

    Y𝑌Y is noetherian and regular, and π𝜋\pi is unramified at the generic points of Y𝑌Y.

Then there exists A[A]superscript𝐴delimited-[]𝐴A^{\prime}\in[A] admitting a λ𝜆\lambda-involution if and only if transfλ([A])=0subscripttransf𝜆delimited-[]𝐴0\operatorname{transf}_{\lambda}([A])=0. In this case, Asuperscript𝐴A^{\prime} can be chosen to have a λ𝜆\lambda-involution of any prescribed coarse type (or any prescribed type, when 222 is invertible on Y𝑌Y) and to satisfy degA=2degAdegsuperscript𝐴2deg𝐴\operatorname{deg}A^{\prime}=2\operatorname{deg}A.

Proof.

This follows from Corollary 6.3.4 and Propositions 6.3.2 and 4.5.3. ∎

Corollary 6.3.6.

Let X𝑋X be a Hausdorff topological space, let λ:XX:𝜆𝑋𝑋\lambda:X\to X be a continuous involution, and let π𝜋\pi denote the quotient map XY:=X/{1,λ}𝑋𝑌assign𝑋1𝜆X\to Y:=X/\{1,\lambda\}. Let A𝐴A be an Azumaya 𝒪Xsubscript𝒪𝑋{\mathcal{O}}_{X}-algebra and let tcTyp(λ)𝑡cTyp𝜆t\in{\mathrm{cTyp}({\lambda})} be a coarse type. Then there exists A[A]superscript𝐴delimited-[]𝐴A^{\prime}\in[A] admitting a λ𝜆\lambda-involution of coarse type t𝑡t if and only if Φ(t)=transfλ([A])Φ𝑡subscripttransf𝜆delimited-[]𝐴\Phi(t)=\operatorname{transf}_{\lambda}([A]) in Hét2(Y,𝒪Y×)subscriptsuperscriptH2ét𝑌superscriptsubscript𝒪𝑌\mathrm{H}^{2}_{\text{\'{e}t}}(Y,{{\mathcal{O}}_{Y}^{\times}}). The algebra Asuperscript𝐴A^{\prime} can be chosen such that degA=2degAdegsuperscript𝐴2deg𝐴\operatorname{deg}A^{\prime}=2\operatorname{deg}A.

Proof.

This is a special case of Theorem 6.3.3, see Examples 4.3.4 and Theorem 4.4.4. ∎

Corollary 6.3.7.

In the situation of Corollary 6.3.6, if λ=id𝜆id\lambda=\mathrm{id}, or λ𝜆\lambda acts freely on X𝑋X, then there exists A[A]superscript𝐴delimited-[]𝐴A^{\prime}\in[A] admitting a λ𝜆\lambda-involution if and only if transfλ([A])=0subscripttransf𝜆delimited-[]𝐴0\operatorname{transf}_{\lambda}([A])=0. In this case Asuperscript𝐴A^{\prime} can be chosen to have a λ𝜆\lambda-involution of any prescribed type and to satisfy degA=2degAdegsuperscript𝐴2deg𝐴\operatorname{deg}A^{\prime}=2\operatorname{deg}A.

Proof.

This follows from Corollary 6.3.6 and Propositions 6.3.2 and 4.5.4. ∎

Remark 6.3.8.

Let R𝑅R be a connected semilocal ring, and let λ:RR:𝜆𝑅𝑅\lambda:R\to R be an involution with fixed ring S𝑆S. When R=S𝑅𝑆R=S or R𝑅R is quadratic étale over S𝑆S, it was observed by Saltman [saltman_azumaya_1978, Th. 4.4] that an Azumaya R𝑅R-algebra A𝐴A that is Brauer equivalent to an algebra with a λ𝜆\lambda-involution already possesses a λ𝜆\lambda-involution. Otherwise said, in this special situation, we can choose A=Asuperscript𝐴𝐴A^{\prime}=A in Corollary 6.3.4.

We do not know whether this statement continues to hold if the assumption that R=S𝑅𝑆R=S or R𝑅R is quadratic étale over S𝑆S is dropped. In this case, the fact that R𝑅R is semilocal implies that two Azumaya algebras of the same degree are isomorphic [ojanguren_71]. With this in hand, Corollary 6.3.4 implies that if A𝐴A is equivalent to an Azumaya R𝑅R-algebra admitting a λ𝜆\lambda-involution, then M2×2(A)subscriptM22𝐴\mathrm{M}_{2\times 2}(A) has a λ𝜆\lambda-involution. The problem therefore reduces to the question of whether the existence of a λ𝜆\lambda-involution on M2×2(A)subscriptM22𝐴\mathrm{M}_{2\times 2}(A) implies the existence of a λ𝜆\lambda-involution on A𝐴A. The same question was asked for arbitrary non-commutative semilocal rings A𝐴A in [first_15, §12], where it was also shown that counterexamples, if any exist, are restricted. In particular, returning to the case of Azumaya algebras, it follows from [first_15, Thm. 7.3] that if degAdeg𝐴\operatorname{deg}A is even, then A𝐴A does posses a λ𝜆\lambda-involution when M2×2(A)subscriptM22𝐴\mathrm{M}_{2\times 2}(A) has one.

6.4. The Kernel of the Transfer Map

We continue to use R𝑅R, S𝑆S, N𝑁N, T𝑇T defined in Subsection 6.3.

Saltman’s theorem can also be regarded as a result characterizing the kernel of the transfer map in terms of existence of certain involutions. We now use Theorem 6.3.3 to generalize this particular aspect, namely, describing the kernel of transfλ:Br(𝐗,𝒪𝐗)H2(𝐘,𝒪𝐘×):subscripttransf𝜆Br𝐗subscript𝒪𝐗superscriptH2𝐘superscriptsubscript𝒪𝐘\operatorname{transf}_{\lambda}:\operatorname{Br}({\mathbf{X}},{\mathcal{O}}_{\mathbf{X}})\to\mathrm{H}^{2}({\mathbf{Y}},{{\mathcal{O}}_{\mathbf{Y}}^{\times}}) in terms of the involutions that the Brauer classes support. For that purpose, we introduce the following families of λ𝜆\lambda-involutions.

Definition 6.4.1.

Let A𝐴A be an Azumaya 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}}-algebra. A λ𝜆\lambda-involution τ:AΛA:𝜏𝐴Λ𝐴\tau:A\to\Lambda A is called semiordinary if there exists a split Azumaya 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}}-algebra Asuperscript𝐴A^{\prime} and a λ𝜆\lambda-involution τ:AΛA:superscript𝜏superscript𝐴Λsuperscript𝐴\tau^{\prime}:A^{\prime}\to\Lambda A^{\prime} such that (πA,πτ)subscript𝜋𝐴subscript𝜋𝜏(\pi_{*}A,\pi_{*}\tau) and (πA,πτ)subscript𝜋superscript𝐴subscript𝜋superscript𝜏(\pi_{*}A^{\prime},\pi_{*}\tau^{\prime}) are locally isomorphic. If Asuperscript𝐴A^{\prime} can moreover be chosen to be Mn×n(𝒪𝐗)subscriptM𝑛𝑛subscript𝒪𝐗\mathrm{M}_{n\times n}({\mathcal{O}}_{\mathbf{X}}) and there is aH0(A)𝑎superscriptH0superscript𝐴a\in\mathrm{H}^{0}(A^{\prime}) such that τsuperscript𝜏\tau^{\prime} is given by x(axa1)λtrmaps-to𝑥superscript𝑎𝑥superscript𝑎1𝜆trx\mapsto(axa^{-1})^{\lambda\text{\rm tr}} on sections, we say that τ𝜏\tau is ordinary.

When τ𝜏\tau is not semiordinary, we shall say it is extraordinary.

Theorem 6.4.2.

With notation as in Theorem 6.3.3, let A𝐴A be an Azumaya 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}}-algebra of degree n𝑛n. Then the following conditions are equivalent:

  1. (a)

    transfλ([A])=0subscripttransf𝜆delimited-[]𝐴0\operatorname{transf}_{\lambda}([A])=0,

  2. (b)

    there exists A[A]superscript𝐴delimited-[]𝐴A^{\prime}\in[A] admitting a semiordinary λ𝜆\lambda-involution,

  3. (c)

    there exists A[A]superscript𝐴delimited-[]𝐴A^{\prime}\in[A] admitting a ordinary λ𝜆\lambda-involution.

In (b), the algebra Asuperscript𝐴A^{\prime} can be chosen to satisfy degA=2degAdegsuperscript𝐴2deg𝐴\operatorname{deg}A^{\prime}=2\operatorname{deg}A and to have a semiordinary involution of any prescribed coarse type in ker(Φ:H0(T)H2(S×))kernel:ΦsuperscriptH0𝑇superscriptH2superscript𝑆\ker(\Phi:\mathrm{H}^{0}(T)\to\mathrm{H}^{2}({S^{\times}})). In (c), the algebra Asuperscript𝐴A^{\prime} can be chosen to satisfy degA=2degAdegsuperscript𝐴2deg𝐴\operatorname{deg}A^{\prime}=2\operatorname{deg}A and to have an ordinary involution of any prescribed coarse type in im(H0(N)H0(T))imsuperscriptH0𝑁superscriptH0𝑇\operatorname{im}(\mathrm{H}^{0}(N)\to\mathrm{H}^{0}(T)).

We shall see below (Corollary 6.4.7) that in the situation of a scheme on which 222 is invertible and a trivial involution, or a quadratic étale covering of schemes with its canonical involution, all λ𝜆\lambda-involutions are ordinary. Thus, Theorem 6.4.2 recovers Saltman’s Theorem when 222 is invertible.

More generally, it will turn out that under mild assumptions, all involutions are ordinary when π:𝐗𝐘:𝜋𝐗𝐘\pi:{\mathbf{X}}\to{\mathbf{Y}} is unramified or everywhere ramified.

Proof.

As in the proof of Theorem 6.3.3, we switch to Azumaya R𝑅R-algebras by applying πsubscript𝜋\pi_{*}.

(c)\implies(b) is clear.

(b)\implies(a): Suppose A[A]superscript𝐴delimited-[]𝐴A^{\prime}\in[A] admits a semiordinary involution τ𝜏\tau and let (B,θ)𝐵𝜃(B,\theta) be a split Azumaya R𝑅R-algebra with a λ𝜆\lambda-involution that is locally isomorphic to (A,τ)superscript𝐴𝜏(A^{\prime},\tau). Then by Theorem 6.3.3 and Proposition 5.2.9, transfλ([A])=Φ(ct(τ))=Φ(ct(θ))=transfλ([B])=0subscripttransf𝜆delimited-[]𝐴Φct𝜏Φct𝜃subscripttransf𝜆delimited-[]𝐵0\operatorname{transf}_{\lambda}([A])=\Phi(\mathrm{ct}(\tau))=\Phi(\mathrm{ct}(\theta))=\operatorname{transf}_{\lambda}([B])=0.

(a)\implies(c): Let tim(H0(N)H0(T))𝑡imsuperscriptH0𝑁superscriptH0𝑇t\in\operatorname{im}(\mathrm{H}^{0}(N)\to\mathrm{H}^{0}(T)). Then t𝑡t is the image of some εH0(N)𝜀superscriptH0𝑁\varepsilon\in\mathrm{H}^{0}(N). We revisit the proof of the “if” part in Theorem 6.3.3 and apply it with our t𝑡t and ε𝜀\varepsilon to obtain an Azumaya R𝑅R-algebra with involution (A,τ)superscript𝐴𝜏(A^{\prime},\tau) such that A[A]superscript𝐴delimited-[]𝐴A^{\prime}\in[A], degA=2ndegsuperscript𝐴2𝑛\operatorname{deg}A^{\prime}=2n, ct(τ)=tct𝜏𝑡\mathrm{ct}(\tau)=t and (AV0,τV0)subscriptsuperscript𝐴subscript𝑉0subscript𝜏subscript𝑉0(A^{\prime}_{V_{0}},\tau_{V_{0}}) is isomorphic to (M2n×2n(RV0),σ)subscriptM2𝑛2𝑛subscript𝑅subscript𝑉0𝜎(\mathrm{M}_{2n\times 2n}(R_{V_{0}}),\sigma) with σ𝜎\sigma being given by x([01ε0]x[01ε0]1)λtrmaps-to𝑥superscriptdelimited-[]01𝜀0𝑥superscriptdelimited-[]01𝜀01𝜆trx\mapsto([\begin{smallmatrix}0&1\\ \varepsilon&0\end{smallmatrix}]x[\begin{smallmatrix}0&1\\ \varepsilon&0\end{smallmatrix}]^{-1})^{\lambda\text{\rm tr}} on sections. Since εH0(N)𝜀superscriptH0𝑁\varepsilon\in\mathrm{H}^{0}(N), the involution σ𝜎\sigma descends to an involution on M2n×2n(R)subscriptM2𝑛2𝑛𝑅\mathrm{M}_{2n\times 2n}(R), defined by the same formula as σ𝜎\sigma, hence τ𝜏\tau is ordinary.

It remains to show that we can choose Asuperscript𝐴A^{\prime} to have a semiordinary involution with a prescribed coarse type tkerΦ𝑡kernelΦt\in\ker\Phi. Let V𝑉V\to\ast be a covering such that t𝑡t lifts to some εN(V)𝜀𝑁𝑉\varepsilon\in N(V). Again, we apply the proof of the “if” part of Theorem 6.3.3 with t𝑡t, ε𝜀\varepsilon and A𝐴A to obtain an Azumaya R𝑅R-algebra with involution (A,τ)superscript𝐴𝜏(A^{\prime},\tau) satisfying A[A]superscript𝐴delimited-[]𝐴A^{\prime}\in[A] and ct(τ)=tct𝜏𝑡\mathrm{ct}(\tau)=t. We then reapply the proof with Mn×n(R)subscriptM𝑛𝑛𝑅\mathrm{M}_{n\times n}(R) in place of A𝐴A to obtain another Azumaya R𝑅R-algebra with involution (A1,τ1)subscriptsuperscript𝐴1subscript𝜏1(A^{\prime}_{1},\tau_{1}) such that A1subscriptsuperscript𝐴1A^{\prime}_{1} is split and ct(τ1)=tctsubscript𝜏1𝑡\mathrm{ct}(\tau_{1})=t. By construction, after suitable refinement of V𝑉V\to*, both (AV,τV)subscriptsuperscript𝐴𝑉subscript𝜏𝑉(A^{\prime}_{V},\tau_{V}) and (A1,V,τ1,V)subscriptsuperscript𝐴1𝑉subscript𝜏1𝑉(A^{\prime}_{1,V},\tau_{1,V}) are isomorphic to (M2n×2n(RV),σ)subscriptM2𝑛2𝑛subscript𝑅𝑉𝜎(\mathrm{M}_{2n\times 2n}(R_{V}),\sigma), where is σ𝜎\sigma given by x([01ε0]x[01ε0]1)λtrmaps-to𝑥superscriptdelimited-[]01𝜀0𝑥superscriptdelimited-[]01𝜀01𝜆trx\mapsto([\begin{smallmatrix}0&1\\ \varepsilon&0\end{smallmatrix}]x[\begin{smallmatrix}0&1\\ \varepsilon&0\end{smallmatrix}]^{-1})^{\lambda\text{\rm tr}} on sections. Consequently, (A,τ)superscript𝐴𝜏(A^{\prime},\tau) and (A1,τ1)subscriptsuperscript𝐴1subscript𝜏1(A^{\prime}_{1},\tau_{1}) are locally isomorphic and therefore τ𝜏\tau is semiordinary. ∎

We now shift our attention from the involutions τ𝜏\tau to the coarse types t𝑡t.

Definition 6.4.3.

Let tH0(T)𝑡superscriptH0𝑇t\in\mathrm{H}^{0}(T) be a coarse λ𝜆\lambda-type. We say that t𝑡t is realizable if there exists some Azumaya 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}}-algebra A𝐴A and some λ𝜆\lambda-involution τ:AΛA:𝜏𝐴Λ𝐴\tau:A\to\Lambda A with coarse type t𝑡t. We also say that t𝑡t is realizable in degree n𝑛n when A𝐴A can be chosen so that n=degA𝑛deg𝐴n=\operatorname{deg}A. When τ𝜏\tau can be chosen to be ordinary, resp. semiordinary, we call t𝑡t ordinary, resp. semiordinary.

The following theorem characterizes the realizable, semiordinary, and ordinary coarse types in cohomological terms.

Theorem 6.4.4.

With notation as in Theorem 6.3.3, let tH0(T)𝑡superscriptH0𝑇t\in\mathrm{H}^{0}(T) be coarse type and let δ0:H0(T)H1(R×/S×):superscript𝛿0superscriptH0𝑇superscriptH1superscript𝑅superscript𝑆\delta^{0}:\mathrm{H}^{0}(T)\to\mathrm{H}^{1}({R^{\times}}/{S^{\times}}), δ1:H1(R×/S×)H2(S×):superscript𝛿1superscriptH1superscript𝑅superscript𝑆superscriptH2superscript𝑆\delta^{1}:\mathrm{H}^{1}({R^{\times}}/{S^{\times}})\to\mathrm{H}^{2}({S^{\times}}) and Φ=δ1δ0Φsuperscript𝛿1superscript𝛿0\Phi=\delta^{1}\circ\delta^{0} be as in Subsection 6.3. Then:

  1. (i)

    t𝑡t is realizable if and only if Φ(t)im(transfλ:Br(𝐗,𝒪𝐗)H2(𝐘,𝒪𝐘×))Φ𝑡im:subscripttransf𝜆Br𝐗subscript𝒪𝐗superscriptH2𝐘superscriptsubscript𝒪𝐘\Phi(t)\in\operatorname{im}(\operatorname{transf}_{\lambda}:\operatorname{Br}({\mathbf{X}},{\mathcal{O}}_{\mathbf{X}})\to\mathrm{H}^{2}({\mathbf{Y}},{{\mathcal{O}}_{\mathbf{Y}}^{\times}})).

  2. (ii)

    t𝑡t is semiordinary if and only if Φ(t)=δ1δ0(t)=0Φ𝑡superscript𝛿1superscript𝛿0𝑡0\Phi(t)=\delta^{1}\delta^{0}(t)=0.

  3. (iii)

    t𝑡t is ordinary if and only if δ0(t)=0superscript𝛿0𝑡0\delta^{0}(t)=0, or equivalently tim(H0(N)H0(T))𝑡imsuperscriptH0𝑁superscriptH0𝑇t\in\operatorname{im}(\mathrm{H}^{0}(N)\to\mathrm{H}^{0}(T)).

When (ii) or (iii) hold, t𝑡t is realizable in degree 222, and hence in all even degrees.

Proof.
  1. (i)

    This follows form Theorem 6.3.3.

  2. (ii)

    The “only if” part follows from Theorems 6.4.2 and 6.3.3. The “if” part and the assertion that t𝑡t can be realized in degree 222 follow by applying Theorem 6.4.2 with A=R𝐴𝑅A=R.

  3. (iii)

    Suppose t𝑡t is ordinary, say t=ct(Mn(R),τ)𝑡ctsubscriptM𝑛𝑅𝜏t=\mathrm{ct}(\mathrm{M}_{n}(R),\tau) with τ𝜏\tau given by x(hxh1)λtrmaps-to𝑥superscript𝑥superscript1𝜆trx\mapsto(hxh^{-1})^{\lambda\text{\rm tr}} on sections. Then we can apply Construction 5.2.5 by with U=𝑈U=*, ψ=id𝜓id\psi=\mathrm{id}, and hh as above, resulting in εH0(N)𝜀superscriptH0𝑁\varepsilon\in\mathrm{H}^{0}(N), which then maps onto tH0(T)𝑡superscriptH0𝑇t\in\mathrm{H}^{0}(T).

    The reverse implication follows by applying Theorem 6.4.2 with A=R𝐴𝑅A=R. ∎

Corollary 6.4.5.

With the notation of Theorem 6.3.3, suppose 𝒪𝐘×superscriptsubscript𝒪𝐘{{\mathcal{O}}_{\mathbf{Y}}^{\times}} has square roots locally, and assume further that 2𝒪𝐘×2superscriptsubscript𝒪𝐘2\in{{\mathcal{O}}_{\mathbf{Y}}^{\times}} or π:𝐗𝐘:𝜋𝐗𝐘\pi:{\mathbf{X}}\to{\mathbf{Y}} is unramified. Let (A,τ)𝐴𝜏(A,\tau) be an Azumaya 𝒪𝐗subscript𝒪𝐗{\mathcal{O}}_{\mathbf{X}}-algebra with a λ𝜆\lambda-involution. Then τ𝜏\tau is ordinary, resp. semiordinary, if and only if its coarse type is.

Proof.

The “only if” part is clear, so we turn to the “if” part. We replace (A,τ)𝐴𝜏(A,\tau) with (πA,πτ)subscript𝜋𝐴subscript𝜋𝜏(\pi_{*}A,\pi_{*}\tau), see Theorem 4.3.11 and Corollary 4.3.14, and write t=ct(τ)𝑡ct𝜏t=\mathrm{ct}(\tau). In case 𝐘𝐘{\mathbf{Y}} is not connected, we express 𝐘subscript𝐘*_{\mathbf{Y}} as nYnsubscriptsquare-union𝑛subscript𝑌𝑛\bigsqcup_{n\in\mathbb{N}}Y_{n} such that AYnsubscript𝐴subscript𝑌𝑛A_{Y_{n}} has degree n𝑛n, and work with each component separately. We may therefore assume that n:=degAassign𝑛deg𝐴n:=\operatorname{deg}A is constant.

By Theorem 5.2.13, it is enough to find an Azumaya R𝑅R-algebra Asuperscript𝐴A^{\prime} with an ordinary, resp. semiordinary, involution τsuperscript𝜏\tau^{\prime} such that degA=degAdeg𝐴degsuperscript𝐴\operatorname{deg}A=\operatorname{deg}A^{\prime} and ct(τ)=ct(τ)ct𝜏ctsuperscript𝜏\mathrm{ct}(\tau)=\mathrm{ct}(\tau^{\prime}). If n𝑛n is odd, then t=1𝑡1t=1 by Theorem 5.2.17(iii), and we can take (A,τ)=(Mn×n(R),λtr)superscript𝐴superscript𝜏subscriptM𝑛𝑛𝑅𝜆tr(A^{\prime},\tau^{\prime})=(\mathrm{M}_{n\times n}(R),\lambda\text{\rm tr}). Otherwise, n=2m𝑛2𝑚n=2m, and applying Theorem 6.4.2 to Mm×m(R)subscriptM𝑚𝑚𝑅\mathrm{M}_{m\times m}(R) yields an algebra with an ordinary, resp. semiordinary, involution (A,τ)superscript𝐴superscript𝜏(A^{\prime},\tau^{\prime}) such that ct(τ)=ct(τ)ctsuperscript𝜏ct𝜏\mathrm{ct}(\tau^{\prime})=\mathrm{ct}(\tau) and degA=degAdegsuperscript𝐴deg𝐴\operatorname{deg}A^{\prime}=\operatorname{deg}A; here we used parts (ii) and (iii) of Theorem 6.4.4. ∎

Corollary 6.4.6.

With the notation of Theorem 6.3.3, suppose that

  1. (1)

    π:𝐗𝐘:𝜋𝐗𝐘\pi:{\mathbf{X}}\to{\mathbf{Y}} is a trivial quotient (Example 4.3.5), or

  2. (2)

    π:𝐗𝐘:𝜋𝐗𝐘\pi:{\mathbf{X}}\to{\mathbf{Y}} is everywhere ramified, 2𝒪𝐘×2superscriptsubscript𝒪𝐘2\in{{\mathcal{O}}_{\mathbf{Y}}^{\times}} and 𝒪𝐘×superscriptsubscript𝒪𝐘{{\mathcal{O}}_{\mathbf{Y}}^{\times}} has square roots locally, or

  3. (3)

    π:𝐗𝐘:𝜋𝐗𝐘\pi:{\mathbf{X}}\to{\mathbf{Y}} is unramified.

Then all coarse λ𝜆\lambda-types are realizable and ordinary.

Proof.

It follows from the proof of Proposition 6.3.2 that in all three cases, δ0:H0(T)H0(R×/S×):superscript𝛿0superscriptH0𝑇superscriptH0superscript𝑅superscript𝑆\delta^{0}:\mathrm{H}^{0}(T)\to\mathrm{H}^{0}({R^{\times}}/{S^{\times}}) is the 00 map. Now apply Theorem 6.4.4(iii). ∎

Corollary 6.4.7.

With the notation of Theorem 6.3.3, suppose that 𝒪𝐘×superscriptsubscript𝒪𝐘{{\mathcal{O}}_{\mathbf{Y}}^{\times}} has square roots locally and moreover

  1. (1)

    π:𝐗𝐘:𝜋𝐗𝐘\pi:{\mathbf{X}}\to{\mathbf{Y}} is everywhere ramified and 2𝒪𝐘×2superscriptsubscript𝒪𝐘2\in{{\mathcal{O}}_{\mathbf{Y}}^{\times}}, or

  2. (2)

    π:𝐗𝐘:𝜋𝐗𝐘\pi:{\mathbf{X}}\to{\mathbf{Y}} is unramified.

Then all λ𝜆\lambda-involutions are ordinary.

Proof.

This follows from Corollaries 6.4.5 and 6.4.6. ∎

Corollary 6.4.8.

Let X𝑋X be a scheme, let λ:XX:𝜆𝑋𝑋\lambda:X\to X be an involution, and let π:XY:𝜋𝑋𝑌\pi:X\to Y be a good quotient relative to {1,λ}1𝜆\{1,\lambda\}. Assume Y𝑌Y is noetherian and regular, and π𝜋\pi is quadratic étale on the generic points of Y𝑌Y. Then all coarse λ𝜆\lambda-types are realizable and semiordinary. If moreover 222 is invertible on Y𝑌Y, then all λ𝜆\lambda-involutions are semiordinary.

Proof.

The first assertion follows from Proposition 6.3.2(iv) and Theorem 6.4.4(ii). The second assertion then follows from Corollary 6.4.5. ∎

We conclude this section with two problems, both of which are open both in the context of varieties over fields of characteristic different from 222 with (ramified) involutions and in the context of topological spaces with (non-free) C2subscript𝐶2C_{2}-actions.

Problem 6.4.9.

Is there an element tcTyp(λ)=H0(T)𝑡cTyp𝜆superscriptH0𝑇t\in{\mathrm{cTyp}({\lambda})}=\mathrm{H}^{0}(T) that is not the coarse type of any Azumaya algebra with λ𝜆\lambda-involution?

Problem 6.4.10.

Is there an Azumaya algebra A𝐴A with a λ𝜆\lambda-involution τ𝜏\tau that is extraordinary (i.e. not semiordinary)?

By Theorem 5.4.5, the first problem can be phrased as follows: Suppse that X𝑋X is a scheme with involution λ𝜆\lambda admitting a good quotient relative to {1,λ}1𝜆\{1,\lambda\}, or X𝑋X is a Hausdorff topological space with involution λ𝜆\lambda. Let Z𝑍Z be the locus of points where λ𝜆\lambda ramifies and let Z=Z1Z1𝑍square-unionsubscript𝑍1subscript𝑍1Z=Z_{-1}\sqcup Z_{1} be a partition of Z𝑍Z into two closed subsets. Is it always possible to find an Azumaya algebra A𝐴A over X𝑋X admitting a λ𝜆\lambda-involution τ𝜏\tau such that the specialization of τ𝜏\tau to k(z)𝑘𝑧k(z) is orthogonal if zZ1𝑧subscript𝑍1z\in Z_{1} and symplectic if zZ1𝑧subscript𝑍1z\in Z_{-1}?

7. Examples and Applications

Example 7.1.1.

Fix an exact quotient π:(𝐗,𝒪𝐗)(𝐘,𝒪𝐘):𝜋𝐗subscript𝒪𝐗𝐘subscript𝒪𝐘\pi:(\mathbf{X},{\mathcal{O}}_{\mathbf{X}})\to(\mathbf{Y},{\mathcal{O}}_{\mathbf{Y}}) and write R=π𝒪𝐗𝑅subscript𝜋subscript𝒪𝐗R=\pi_{*}\mathcal{O}_{\mathbf{X}}, S=𝒪𝐘𝑆subscript𝒪𝐘S={\mathcal{O}}_{\mathbf{Y}}. We assume that S×superscript𝑆{S^{\times}} has square roots locally and 2S×2superscript𝑆2\in{S^{\times}}. This assumption allows us to drop the distinction between types and coarse types for the most part (Corollary 5.2.14). As in the previous section, we use the notation N𝑁N for the kernel the λ𝜆\lambda-norm map xxλx:R×S×:maps-to𝑥superscript𝑥𝜆𝑥superscript𝑅superscript𝑆{x\mapsto x^{\lambda}x}:{R^{\times}}\to{S^{\times}}, and T𝑇T for the quotient of N𝑁N by the image of the map R×Nsuperscript𝑅𝑁{R^{\times}}\to N given by rr1rλmaps-to𝑟superscript𝑟1superscript𝑟𝜆r\mapsto r^{-1}r^{\lambda}. The coarse types are then H0(𝐘,T)superscriptH0𝐘𝑇\mathrm{H}^{0}(\mathbf{Y},T).

Suppose t𝑡t is an ordinary coarse type. By Theorem 6.4.4, this is equivalent to saying there exists some ε𝜀\varepsilon in H0(𝐘,N)superscriptH0𝐘𝑁\mathrm{H}^{0}(\mathbf{Y},N) mapping to t𝑡t under the map H0(𝐘,N)H0(𝐘,T)superscriptH0𝐘𝑁superscriptH0𝐘𝑇\mathrm{H}^{0}(\mathbf{Y},N)\to\mathrm{H}^{0}(\mathbf{Y},T). Such an ε𝜀\varepsilon can always be found if H1(𝐘,R×/S×)superscriptH1𝐘superscript𝑅superscript𝑆\mathrm{H}^{1}(\mathbf{Y},{R^{\times}}/{S^{\times}}) vanishes, for instance.

Let n𝑛n be a natural number. Consider the matrix

h=h2n(ε)=[0InεIn0].subscript2𝑛𝜀matrix0subscript𝐼𝑛𝜀subscript𝐼𝑛0h=h_{2n}(\varepsilon)=\begin{bmatrix}0&I_{n}\\ \varepsilon I_{n}&0\end{bmatrix}.

It is immediate that εhλtr=h𝜀superscript𝜆tr\varepsilon h^{\lambda\text{\rm tr}}=h. This equality implies that the map τε:Mat2n×2n(R)Mat2n×2n(R):subscript𝜏𝜀subscriptMat2𝑛2𝑛𝑅subscriptMat2𝑛2𝑛𝑅\tau_{\varepsilon}:\operatorname{Mat}_{2n\times 2n}(R)\to\operatorname{Mat}_{2n\times 2n}(R) given on sections by

M(h2n(ε)Mh2n(ε)1)λtr.maps-to𝑀superscriptsubscript2𝑛𝜀𝑀subscript2𝑛superscript𝜀1𝜆trM\mapsto(h_{2n}(\varepsilon)\,M\,h_{2n}(\varepsilon)^{-1})^{\lambda\text{\rm tr}}.

is a λ𝜆\lambda-involution. The (coarse) type of τ𝜏\tau is easily seen to be t𝑡t, the image of ε𝜀\varepsilon in H0(T)superscriptH0𝑇\mathrm{H}^{0}(T). This follows from Construction 5.2.5.

In this case, any algebra of degree 2n2𝑛2n with involution of coarse type t𝑡t is locally isomorphic to (Mat2n×2n(R),τ)subscriptMat2𝑛2𝑛𝑅𝜏(\operatorname{Mat}_{2n\times 2n}(R),\tau), by Theorem 5.2.13. Thanks to Corollary 5.2.19, we may therefore place such algebras in bijective correspondence with G𝐺G-torsors on 𝐘𝐘\mathbf{Y} where G=PU(Mat2n×2n(R),τε)𝒜utR(Mat2n×2n(R),τε)𝐺PUsubscriptMat2𝑛2𝑛𝑅subscript𝜏𝜀𝒜𝑢subscript𝑡𝑅subscriptMat2𝑛2𝑛𝑅subscript𝜏𝜀G=\operatorname{PU}(\operatorname{Mat}_{2n\times 2n}(R),\tau_{\varepsilon})\cong\mathcal{A}ut_{R}(\operatorname{Mat}_{2n\times 2n}(R),\tau_{\varepsilon}).

Example 7.1.2.

As a special case of the previous example, we describe the Azumaya algebras with symplectic involution on a scheme or topological space with trivial involution. See Theorem 4.4.4 for the specific hypotheses on the underlying geometric object, and note that we assume 222 is invertible.

In this case, π=id𝜋id\pi=\mathrm{id}, R=S𝑅𝑆R=S, and N=μ2,R𝑁subscript𝜇2𝑅N=\mu_{2,R}. By Theorem 5.2.17(i), the group of coarse types is H0(μ2,R)superscriptH0subscript𝜇2𝑅\mathrm{H}^{0}(\mu_{2,R}), which is just {1,1}11\{1,-1\} when X𝑋X is connected. We consider the (coarse) type 11-1, called the symplectic type.

Any Azumaya algebra with involution having this type must be of even degree, 2n2𝑛2n, by Theorem 5.2.17(iii), and is locally isomorphic to the split degree-2n2𝑛2n algebra with symplectic involution

sp:M(h2n(1)Mh2n(1)1)tr.:spmaps-to𝑀superscriptsubscript2𝑛1𝑀subscript2𝑛superscript11tr{\mathrm{sp}}:M\mapsto(h_{2n}(-1)Mh_{2n}(-1)^{-1})^{\text{\rm tr}}.

The unitary group of (M2n×2n(R),sp)subscriptM2𝑛2𝑛𝑅sp(\mathrm{M}_{2n\times 2n}(R),{\mathrm{sp}}) is the familiar symplectic group Sp2n(R)subscriptSp2𝑛𝑅{\mathrm{Sp}}_{2n}(R), and it follows from Lemma 5.2.18 that the automorphism group of (M2n×2n(R),sp)subscriptM2𝑛2𝑛𝑅sp(\mathrm{M}_{2n\times 2n}(R),{\mathrm{sp}}) is

PSp2n(R):=Spn(R)/μ2,R.assignsubscriptPSp2𝑛𝑅subscriptSp𝑛𝑅subscript𝜇2𝑅\operatorname{PSp}_{2n}(R):={\mathrm{Sp}}_{n}(R)/\mu_{2,R}\ .

In particular, as noted in Corollary 5.2.19, the set of isomorphism classes of degree-2n2𝑛2n Azumaya algebras with symplectic involution is in canonical bijection with

H1(𝐗,PSp2n(R)).superscriptH1𝐗subscriptPSp2𝑛𝑅\mathrm{H}^{1}(\mathbf{X},\operatorname{PSp}_{2n}(R)).

Since the symplectic type is ordinary, by Theorem 6.4.2 and Example 6.2.2, we derive the well known fact that an Azumaya algebra A𝐴A on X𝑋X is Brauer equivalent to one having a symplectic involution if and only if the Brauer class of A𝐴A is 222-torsion.

Example 7.1.3.

Fix an exact quotient π:𝐗𝐘:𝜋𝐗𝐘\pi:\mathbf{X}\to\mathbf{Y} with ring objects 𝒪𝐗subscript𝒪𝐗\mathcal{O}_{\mathbf{X}}, and let R=π𝒪𝐗𝑅subscript𝜋subscript𝒪𝐗R=\pi_{*}\mathcal{O}_{\mathbf{X}} and S=𝒪𝐘𝑆subscript𝒪𝐘S=\mathcal{O}_{{\mathbf{Y}}}. Let n𝑛n be a natural number and assume that the hypotheses of Theorem 5.2.13 hold, namely S×superscript𝑆{S^{\times}} has square roots locally, and either 2S×2superscript𝑆2\in{S^{\times}}, or π𝜋\pi is unramified, or n𝑛n is odd. We consider the trivial type, 111. This is the type of the involution

MMλtrmaps-to𝑀superscript𝑀𝜆trM\mapsto M^{\lambda\text{\rm tr}}

on the split algebra Matn×n(R)subscriptMat𝑛𝑛𝑅\operatorname{Mat}_{n\times n}(R).

Any algebra with involution of the trivial type is locally isomorphic to this one, and therefore, as summarized in Corollary 5.2.19, these are classified by G𝐺G torsors where G=𝒜utR(Mn×n(R),λtr)PU(Mn×n(R),λtr)𝐺𝒜𝑢subscript𝑡𝑅subscriptM𝑛𝑛𝑅𝜆trPUsubscriptM𝑛𝑛𝑅𝜆trG=\mathcal{A}ut_{R}(\mathrm{M}_{n\times n}(R),\lambda\text{\rm tr})\cong\operatorname{PU}(\mathrm{M}_{n\times n}(R),\lambda\text{\rm tr}). We write the latter group as

PUn(R,λ)subscriptPU𝑛𝑅𝜆\operatorname{PU}_{n}(R,\lambda)

and call it the projective unitary group of rank n𝑛n for the involution λ𝜆\lambda. In accordance with this notation, the unitary group of (Mn×n(R),λtr)subscriptM𝑛𝑛𝑅𝜆tr(\mathrm{M}_{n\times n}(R),\lambda\text{\rm tr}) will be denoted Un(R,λ)subscriptU𝑛𝑅𝜆\operatorname{U}_{n}(R,\lambda).

Example 7.1.4.

Consider the case of a scheme or a topological space X𝑋X with trivial involution, as in the case of Example 7.1.2. The theory of Azumaya algebras with involution of type 111 can be established along the same lines as that of type 11-1. These algebras are called orthogonal. The automorphism group of (M2n×2n(R),tr)subscriptM2𝑛2𝑛𝑅tr(\mathrm{M}_{2n\times 2n}(R),\text{\rm tr}) is the quotient group O2n(R)/μ2,Rsubscript𝑂2𝑛𝑅subscript𝜇2𝑅O_{2n}(R)/\mu_{2,R}, which we denote by PO2n(R)subscriptPO2𝑛𝑅\operatorname{PO}_{2n}(R), the projective orthogonal group. This is special notation for the group PU2n(R,id)subscriptPU2𝑛𝑅id\operatorname{PU}_{2n}(R,\mathrm{id}) of Example 7.1.3.

Again, by reference to Theorem 6.4.2 and Example 6.2.2, an algebra is Brauer equivalent to one having involution of this type if and only if the Brauer class is 222-torsion.

Example 7.1.5.

In this example we discuss unitary involutions. As a special case of Example 7.1.3, we consider the case of an unramified double covering π:XY:𝜋𝑋𝑌\pi:X\to Y of schemes or topological spaces. Again, we refer to Theorem 4.4.4 for the specific hypotheses on the underlying geometric object.

In this case, the ring object R𝑅R is a quadratic étale extension of S𝑆S, see Propositions 4.5.3 and 4.5.4. Since π𝜋\pi is unramified, Theorem 5.2.17(ii) implies that there is only one type of involution on Azumaya algebras, the trivial one, which is called unitary in this context. In particular, we are in a special case of Example 7.1.3.

The structure of the groups Un(R,λ)subscriptU𝑛𝑅𝜆\operatorname{U}_{n}(R,\lambda) and PUn(R,λ)=Un(R,λ)/NsubscriptPU𝑛𝑅𝜆subscriptU𝑛𝑅𝜆𝑁\operatorname{PU}_{n}(R,\lambda)=\operatorname{U}_{n}(R,\lambda)/N depends on the nature of λ𝜆\lambda, so a complete description in the abstract is not possible. We can, however, find an étale, resp. open, covering UY𝑈𝑌U\to Y such that RUSU×SUsubscript𝑅𝑈subscript𝑆𝑈subscript𝑆𝑈R_{U}\cong S_{U}\times S_{U}. After specializing to U𝑈U, the algebra Mn×n(R)subscriptM𝑛𝑛𝑅\mathrm{M}_{n\times n}(R) becomes Mn×n(S)×Mn×n(S)subscriptM𝑛𝑛𝑆subscriptM𝑛𝑛𝑆\mathrm{M}_{n\times n}(S)\times\mathrm{M}_{n\times n}(S) and the involution λtr𝜆tr\lambda\text{\rm tr} is becomes the involution given sectionwise by (x,y)(ytr,xtr)maps-to𝑥𝑦superscript𝑦trsuperscript𝑥tr(x,y)\mapsto(y^{\text{\rm tr}},x^{\text{\rm tr}}). From this one verifies that Un(R,λ)UGLn(S)U\operatorname{U}_{n}(R,\lambda)_{U}\cong\operatorname{GL}_{n}(S)_{U} and PUn(R,λ)UPGLn(S)U\operatorname{PU}_{n}(R,\lambda)_{U}\cong\operatorname{PGL}_{n}(S)_{U}. Consequently, for a general degree-n𝑛n Azumaya R𝑅R-algebra with involution (A,τ)𝐴𝜏(A,\tau), the groups U(A,τ)U𝐴𝜏\operatorname{U}(A,\tau) and PU(A,τ)PU𝐴𝜏\operatorname{PU}(A,\tau) are locally isomorphic to GLn(S)subscriptGL𝑛𝑆\operatorname{GL}_{n}(S) and PGLn(S)subscriptPGL𝑛𝑆\operatorname{PGL}_{n}(S), respectively.

This agrees with the well established fact that projective unitary group schemes of unitary involutions are of type A𝐴A.

Example 7.1.6.

In another instance of Example 7.1.1, we can produce an example of an Azumaya algebra with an involution that mixes the various classical types. This example also featured in the introduction.

We work with étale sheaves and étale cohomology throughout, see Example 4.3.3. Let k𝑘k be an algebraically closed field and let X=Speck[x,x1]𝑋Spec𝑘𝑥superscript𝑥1X=\operatorname{Spec}k[x,x^{-1}] with the k𝑘k-linear involution λ𝜆\lambda sending x𝑥x to x1superscript𝑥1x^{-1}. The good quotient of X𝑋X by this involution exists, and is given by Y=Speck[y]𝑌Spec𝑘delimited-[]𝑦Y=\operatorname{Spec}k[y] where y=x+x1𝑦𝑥superscript𝑥1y=x+x^{-1}.

Here, the ring object R𝑅R is the ring k[x,x1]𝑘𝑥superscript𝑥1k[x,x^{-1}] viewed as a sheaf of rings on Y𝑌Y, and the ring object S𝑆S is the structure sheaf of Y𝑌Y. The sheaf N𝑁N is the sheaf of norm-111 elements in R𝑅R, where the norm map sends a Laurent polynomial p(x)𝑝𝑥p(x) to p(x)p(x1)𝑝𝑥𝑝superscript𝑥1p(x)p(x^{-1}). Both the Picard and the Brauer groups of X𝑋X and Y𝑌Y vanish, so that we can calculate H1(Y,R×/S×)=0superscriptH1𝑌superscript𝑅superscript𝑆0\mathrm{H}^{1}(Y,{R^{\times}}/{S^{\times}})=0. The following sequence is therefore exact

1H0(Y,R×/S×)𝜓H0(Y,N)H0(Y,T)1.1superscriptH0𝑌superscript𝑅superscript𝑆𝜓superscriptH0𝑌𝑁superscriptH0𝑌𝑇11\to\mathrm{H}^{0}(Y,{R^{\times}}/{S^{\times}})\xrightarrow{\psi}\mathrm{H}^{0}(Y,N)\to\mathrm{H}^{0}(Y,T)\to 1.

Explicitly, we calculate that H0(Y,R×/S×)superscriptH0𝑌superscript𝑅superscript𝑆\mathrm{H}^{0}(Y,{R^{\times}}/{S^{\times}}) consists of classes of monomials xisuperscript𝑥𝑖x^{i} for i𝑖i\in\mathbb{Z}, and H0(Y,N)superscriptH0𝑌𝑁\mathrm{H}^{0}(Y,N) consists of monomials of the form ±xiplus-or-minussuperscript𝑥𝑖\pm x^{i} for i𝑖i\in\mathbb{Z}, but ψ𝜓\psi maps the class of xisuperscript𝑥𝑖x^{i} to xi/xi=x2isuperscript𝑥𝑖superscript𝑥𝑖superscript𝑥2𝑖x^{i}/x^{-i}=x^{2i}. Therefore, the group of (coarse) types is isomorphic to the Klein 444-group: H0(Y,T)={1¯,1¯,x¯,x¯}superscriptH0𝑌𝑇¯1¯1¯𝑥¯𝑥\mathrm{H}^{0}(Y,T)=\{\overline{1},\overline{-1},\overline{x},\overline{-x}\}.

Since H1(Y,R×/S×)=0superscriptH1𝑌superscript𝑅superscript𝑆0\mathrm{H}^{1}(Y,{R^{\times}}/{S^{\times}})=0, we are in the circumstance of Example 7.1.1 which provides models for each of the four types on even-degree split algebras. For instance, on Mat2×2(R)subscriptMat22𝑅\operatorname{Mat}_{{2}\times{2}}(R), we have the involution given by conjugating by [01x0]delimited-[]01𝑥0[\begin{smallmatrix}0&1\\ x&0\end{smallmatrix}] and then applying λtr𝜆tr\lambda\text{\rm tr}, namely

[a(x)b(x)c(x)d(x)][d(x1)x1b(x1)xc(x1)a(x1)].maps-tomatrix𝑎𝑥𝑏𝑥𝑐𝑥𝑑𝑥matrix𝑑superscript𝑥1superscript𝑥1𝑏superscript𝑥1𝑥𝑐superscript𝑥1𝑎superscript𝑥1\begin{bmatrix}a(x)&b(x)\\ c(x)&d(x)\end{bmatrix}\mapsto\begin{bmatrix}d(x^{-1})&x^{-1}b(x^{-1})\\ xc(x^{-1})&a(x^{-1})\end{bmatrix}.

Away from the fixed locus of λ:XX:𝜆𝑋𝑋\lambda:X\to X, namely, the points x=1𝑥1x=1 and x=1𝑥1x=-1, this involution is unitary, whereas at x=1𝑥1x=1 it specializes to be orthogonal and at x=1𝑥1x=-1 to be symplectic.

More generally, it follows from Theorem 5.4.5 that the type of any Azumaya X𝑋X-algebra with involution (A,τ)𝐴𝜏(A,\tau) is determined by the types seen upon specializing to x=1𝑥1x=1 and x=1𝑥1x=-1.

Example 7.1.7.

We now demonstrate that there exist involutions that are not locally isomorphic to involutions of the form exhibited in Example 7.1.1. Specifically, we will show that there are involutions which are not ordinary in the sense of Definition 6.4.1.

We consider a complex hyperelliptic curve X𝑋X of genus g𝑔g and a double covering XY=:1X\to Y=:{\mathbb{P}}^{1}_{\mathbb{C}}. Explicitly: Let a1,,a2g+1subscript𝑎1subscript𝑎2𝑔1a_{1},\dots,a_{2g+1} be distinct complex numbers, and let

X0=Spec[x,y]/(y2i(xai)).subscript𝑋0Spec𝑥𝑦superscript𝑦2subscriptproduct𝑖𝑥subscript𝑎𝑖X_{0}=\operatorname{Spec}\mathbb{C}[x,y]/(y^{2}-\prod_{i}(x-a_{i})).

We complete X0subscript𝑋0X_{0} by gluing it to X1:=Spec[u,v]/(v2ui(1aiu))assignsubscript𝑋1Spec𝑢𝑣superscript𝑣2𝑢subscriptproduct𝑖1subscript𝑎𝑖𝑢X_{1}:=\operatorname{Spec}\mathbb{C}[u,v]/(v^{2}-u\prod_{i}(1-a_{i}u)) by mapping (x,y)𝑥𝑦(x,y) to (u,v):=(1x,yxg+1)assign𝑢𝑣1𝑥𝑦superscript𝑥𝑔1(u,v):=(\frac{1}{x},\frac{y}{x^{g+1}}), and denote the resulting smooth complete curve by X𝑋X. View Y=1𝑌subscriptsuperscript1Y={\mathbb{P}}^{1}_{\mathbb{C}} as the gluing of Y0:=Spec[x]assignsubscript𝑌0Specdelimited-[]𝑥Y_{0}:=\operatorname{Spec}\mathbb{C}[x] to Y1:=Spec[u]assignsubscript𝑌1Specdelimited-[]𝑢Y_{1}:=\operatorname{Spec}\mathbb{C}[u] via (x:1)(1:u1)(x:1)\leftrightarrow(1:u^{-1}). Projection onto the x𝑥x or u𝑢u coordinate induces a double covering π:XY:𝜋𝑋𝑌\pi:X\to Y with ramification at the points (a1:0),,(a2g+1:0)(a_{1}\!:\!0),\dots,(a_{2g+1}\!:\!0) and (0:1):01(0\!:\!1). The map λ:XX:𝜆𝑋𝑋\lambda:X\to X given by (x,y)(x,y)maps-to𝑥𝑦𝑥𝑦(x,y)\mapsto(x,-y), resp. (u,v)(u,v)maps-to𝑢𝑣𝑢𝑣(u,v)\mapsto(u,-v), on the charts is an involution and π𝜋\pi is a good quotient relative to C2:={1,λ}assignsubscript𝐶21𝜆C_{2}:=\{1,\lambda\}. Indeed, working with the affine covering Y=Y0Y1𝑌subscript𝑌0subscript𝑌1Y=Y_{0}\cup Y_{1}, we see that [x]delimited-[]𝑥\mathbb{C}[x] is the fixed ring of

λ#:[x,y](y2i(xai))[x,y](y2i(xai)),xx,yy,:superscript𝜆#formulae-sequence𝑥𝑦superscript𝑦2subscriptproduct𝑖𝑥subscript𝑎𝑖𝑥𝑦superscript𝑦2subscriptproduct𝑖𝑥subscript𝑎𝑖formulae-sequencemaps-to𝑥𝑥maps-to𝑦𝑦\lambda^{\#}:\frac{\mathbb{C}[x,y]}{(y^{2}-\prod_{i}(x-a_{i}))}\to\frac{\mathbb{C}[x,y]}{(y^{2}-\prod_{i}(x-a_{i}))},\qquad x\mapsto x,\quad y\mapsto-y,

and similarly on the other chart.

By Corollary 6.4.8, all coarse λ𝜆\lambda-types in H0(T)superscriptH0𝑇\mathrm{H}^{0}(T) are realizable and semiordinary. Since the branch locus of π𝜋\pi consists of 2g+22𝑔22g+2 points, it follows from Theorem 5.4.5 that there are 22g+2superscript22𝑔22^{2g+2} λ𝜆\lambda-types. Theorem 6.4.4 also says that the number of ordinary types is the cardinality of the image of the map H0(N)H0(T)superscriptH0𝑁superscriptH0𝑇\mathrm{H}^{0}(N)\to\mathrm{H}^{0}(T), where N𝑁N is the sheaf of sections of λ𝜆\lambda-norm 111 in R:=π𝒪Xassign𝑅subscript𝜋subscript𝒪𝑋R:=\pi_{*}{\mathcal{O}}_{X}. Let εH0(N)𝜀superscriptH0𝑁\varepsilon\in\mathrm{H}^{0}(N). Then εH0(Y,π𝒪X)=H0(X,𝒪X)𝜀superscriptH0𝑌subscript𝜋subscript𝒪𝑋superscriptH0𝑋subscript𝒪𝑋\varepsilon\in\mathrm{H}^{0}(Y,\pi_{*}{\mathcal{O}}_{X})=\mathrm{H}^{0}(X,{\mathcal{O}}_{X}). Since X𝑋X is a complete complex curve, the global sections of the structure sheaf are constant functions, meaning that ε×𝜀superscript\varepsilon\in{\mathbb{C}^{\times}}. Since ελε=1superscript𝜀𝜆𝜀1\varepsilon^{\lambda}\varepsilon=1, it follows that H0(N)={±1}superscriptH0𝑁plus-or-minus1\mathrm{H}^{0}(N)=\{\pm 1\}. The images of 1,1H0(N)11superscriptH0𝑁1,-1\in\mathrm{H}^{0}(N) in H0(T)superscriptH0𝑇\mathrm{H}^{0}(T) are therefore the ordinary types. Thus, of the 22g+2superscript22𝑔22^{2g+2} possible coarse types, only 222 are ordinary, and the remaining 22g+22superscript22𝑔222^{2g+2}-2 are merely semiordinary.

We remark that we have reached the latter conclusion without actually constructing Azumaya algebras with involution realizing any of the non-ordinary types. A construction is given in the proof of Theorem 6.4.4, and it can be made explicit in our setting with further work.

This example can also be carried with the affine models X0subscript𝑋0X_{0} and Y0subscript𝑌0Y_{0}. One can check directly that H0(X0,𝒪X0×)=×superscriptH0subscript𝑋0superscriptsubscript𝒪subscript𝑋0superscript\mathrm{H}^{0}(X_{0},{{\mathcal{O}}_{X_{0}}^{\times}})={\mathbb{C}^{\times}} and thus it is still the case that H0(Y,N)={±1}superscriptH0𝑌𝑁plus-or-minus1\mathrm{H}^{0}(Y,N)=\{\pm 1\}. Since the ramification point (0:1)1=Y(0:1)\in{\mathbb{P}}^{1}_{\mathbb{C}}=Y was removed, in this case, there are 22g+1superscript22𝑔12^{2g+1} coarse types, all semiordinary, of which only 222 are ordinary.

Example 7.1.8.

A surprising source of examples comes from Clifford algebras of quadratic forms with simple degeneration. We refer the reader to [auel_fibrations_2014, §1] or [auel_parimala_suresh_2015, §1] for all relevant definitions.

Let Y𝑌Y be a scheme on which 222 is invertible and let (E,q,L)𝐸𝑞𝐿(E,q,L) be a line-bundle-valued quadratic space of even rank n𝑛n over Y𝑌Y; when L=𝒪Y𝐿subscript𝒪𝑌L={\mathcal{O}}_{Y} and Y=SpecS𝑌Spec𝑆Y=\operatorname{Spec}S, these data merely amount to specifying a quadratic space of rank n𝑛n over the ring S𝑆S. According to [auel_parimala_suresh_2015, Dfn. 1.9], q𝑞q is said to have simple degeneration if for every yY𝑦𝑌y\in Y, the specialization of q𝑞q to k(y)𝑘𝑦k(y) is a quadratic form whose radical has dimension at most 111. In this case, it shown in [auel_parimala_suresh_2015, Prp. 1.11] that the even Clifford algebra C0(q)subscript𝐶0𝑞C_{0}(q), which is a sheaf of 𝒪Ysubscript𝒪𝑌{\mathcal{O}}_{Y}-algebras, is Azumaya over its centre Z(q)𝑍𝑞Z(q). Furthermore, the sheaf Z(q)𝑍𝑞Z(q) corresponds to a flat double covering π:XY:𝜋𝑋𝑌\pi:X\to Y, which ramifies at the points yY𝑦𝑌y\in Y where qk(y)subscript𝑞𝑘𝑦q_{k(y)} is degenerate. As such, π𝜋\pi is a good quotient relative to the involution λ:XX:𝜆𝑋𝑋\lambda:X\to X induced by the involution of Z(q)𝑍𝑞Z(q) given by xTrZ(q)/𝒪Y(x)xmaps-to𝑥subscriptTr𝑍𝑞subscript𝒪𝑌𝑥𝑥x\mapsto\mathrm{Tr}_{Z(q)/{\mathcal{O}}_{Y}}(x)-x on sections. Abusing the notation, we realize C0(q)subscript𝐶0𝑞C_{0}(q) as an Azumaya algebra over X𝑋X.

Suppose q𝑞q has simple degeneration. We moreover assume that Y𝑌Y is integral, regular and noetherian with generic point ξ𝜉\xi and that qk(ξ)subscript𝑞𝑘𝜉q_{k(\xi)} is nondegenerate, although it is likely that these assumptions are unnecessary. The algebra C0(q)subscript𝐶0𝑞C_{0}(q) has a canonical involution τ0subscript𝜏0\tau_{0}, see [auel_mpim_2011, §1.8], and by applying [knus_book_1998-1, Prp. 8.4] to C0(qk(ξ))subscript𝐶0subscript𝑞𝑘𝜉C_{0}(q_{k(\xi)}), we see that τ0subscript𝜏0\tau_{0} is of the first kind when n0(mod4)𝑛annotated0pmod4n\equiv 0\pmod{4} and a λ𝜆\lambda-involution when n2(mod4)𝑛annotated2pmod4n\equiv 2\pmod{4}. Furthermore, in the case n0(mod4)𝑛annotated0pmod4n\equiv 0\pmod{4}, we have transfλ([C0(q)])=0subscripttransf𝜆delimited-[]subscript𝐶0𝑞0\operatorname{transf}_{\lambda}([C_{0}(q)])=0 because transfλ([C0(qk(ξ))])=0subscripttransf𝜆delimited-[]subscript𝐶0subscript𝑞𝑘𝜉0\operatorname{transf}_{\lambda}([C_{0}(q_{k(\xi)})])=0 by [knus_book_1998-1, Thm. 9.12], and Br(Y)Br(k(ξ))Br𝑌Br𝑘𝜉\operatorname{Br}(Y)\to\operatorname{Br}(k(\xi)) is injective by [grothendieck_groupe_1968, Cor. 1.8] (or [auslander_brauer_1960, Thm. 7.2] in the affine case). It therefore follows from Theorem 6.3.3 that there exists A[C0(q)]superscript𝐴delimited-[]subscript𝐶0𝑞A^{\prime}\in[C_{0}(q)] with degA=2degC0(q)=2ndegsuperscript𝐴2degsubscript𝐶0𝑞superscript2𝑛\operatorname{deg}A^{\prime}=2\operatorname{deg}C_{0}(q)=2^{n} that admits a λ𝜆\lambda-involution. We expect that the choice of Asuperscript𝐴A^{\prime} and its involution can be done canonically in q𝑞q, and with no restrictions on Y𝑌Y.

With the observations just made, it is possible that our work could facilitate the study of Clifford invariants of non-regular quadratic forms, e.g. in [voight_2011], [auel_fibrations_2014], [auel_parimala_suresh_2015].

8. Topology and Classifying Spaces

The remainder of this paper is concerned with constructing a quadratic étale map of complex varieties XY𝑋𝑌X\to Y and an Azumaya algebra A𝐴A over over X𝑋X such that A𝐴A is Brauer equivalent to an algebra Asuperscript𝐴A^{\prime} with a λ𝜆\lambda-involution, λ𝜆\lambda being the non-trivial Y𝑌Y-automorphism of X𝑋X, but such that the smallest degree of any such Asuperscript𝐴A^{\prime} is 2degA2deg𝐴2\operatorname{deg}A.

We recall that in this particular case, a Brauer equivalent algebra of degree 2degA2deg𝐴2\operatorname{deg}A admitting a λ𝜆\lambda-involution is guaranteed to exist by a theorem of Knus, Parimala and Srinivas [knus_azumaya_1990, Th. 4.2]; this has been generalized in Theorem 6.3.3. An analogous example in which λ:XX:𝜆𝑋𝑋\lambda:X\to X is the trivial involution was exhibited in [asher_auel_azumaya_2017].

The example, which is constructed in Section 9, will be obtained by means of topological obstruction theory, similarly to the methods of [antieau_unramified_2014], [asher_auel_azumaya_2017] and related works. That is, the desired properties of A𝐴A above will be verified by establishing them for the topological Azumaya algebra A()𝐴A(\mathbb{C}) over the complexification X()𝑋X(\mathbb{C}), whereas the latter will be done by means of certain homotopy invariants.

This section is foundational, describing in part an approach to topological Azumaya algebras with involution via equivariant homotopy theory. The main points are that Azumaya algebras with involution correspond to principal PGLn()subscriptPGL𝑛\operatorname{PGL}_{n}(\mathbb{C})-bundles with involution—a fact that is true even outside the topological context, but that we have not emphasized until now—, that there are equivariant classifying spaces for such bundles, and that their theory is tractable if one restricts to considering spaces X𝑋X on which the C2subscript𝐶2C_{2}-action is trivial or free.

8.1. Preliminaries

In this section and the next, all topological spaces will be tacitly assumed to have a number of desirable properties. All spaces appearing will be assumed to be compactly generated, Hausdorff, paracompact and locally contractible.

Throughout, we work in the category of C2subscript𝐶2C_{2}-topological spaces and C2subscript𝐶2C_{2}-equivariant maps. There are two notions of homotopy one can consider for maps in this setting, the fine, in which homotopies are themselves required to be C2subscript𝐶2C_{2}-equivariant, and the coarse, where non-equivariant homotopies are allowed. These two notions each have model structures appropriate to them, the fine and the coarse. In the fine model structure, the weak equivalences are the equivariant maps f:XY:𝑓𝑋𝑌f:X\to Y inducing weak equivalences on fixed point sets f:XGYG:𝑓superscript𝑋𝐺superscript𝑌𝐺f:X^{G}\to Y^{G} where G𝐺G is either the group C2subscript𝐶2C_{2} or the subgroup {1}1\{1\}. In the coarse structure, it is required only that f:XY:𝑓𝑋𝑌f:X\to Y be a weak equivalence when the C2subscript𝐶2C_{2}-action is disregarded, that is, only the subgroup {1}1\{1\} is considered. The identity functor is a left Quillen functor from the coarse to the fine. This is a synthesis of the theory of [dwyer_singular_1984] with [elmendorf_systems_1983].

Notation 8.1.1.

The notation [X,Y]𝑋𝑌[X,Y] is used to denote the set of maps between two (possibly unpointed) objects X𝑋X and Y𝑌Y in a homotopy category. The notation [X,Y]C2subscript𝑋𝑌subscript𝐶2[X,Y]_{C_{2}} will be used to denote the set of maps between X𝑋X and Y𝑌Y in the fine C2subscript𝐶2C_{2}-equivariant homotopy category, whereas [X,Y]C2-coarsesubscript𝑋𝑌subscript𝐶2-coarse[X,Y]_{C_{2}\text{-coarse}} will be used for the coarse structure.

Remark 8.1.2.

In the case of the coarse model structure, the cofibrant objects include the C2subscript𝐶2C_{2}-CW-complexes with free C2subscript𝐶2C_{2}-action, and if X𝑋X is a C2subscript𝐶2C_{2}-CW-complex, then the construction X×EC2X𝑋𝐸subscript𝐶2𝑋X\times EC_{2}\to X furnishes a cofibrant replacement of X𝑋X.

All spaces are fibrant in both the coarse and the fine model structures, which implies the following standard result.

Proposition 8.1.3.

If X𝑋X is a free C2subscript𝐶2C_{2}-CW-complex and Y𝑌Y is a C2subscript𝐶2C_{2}-space, then there is a natural bijection

[X,Y]C2[X,Y]C2-coarse.subscript𝑋𝑌subscript𝐶2subscript𝑋𝑌subscript𝐶2-coarse[X,Y]_{C_{2}}\longleftrightarrow[X,Y]_{C_{2}\text{-coarse}}.

It is well known that C2subscript𝐶2C_{2}-equivariant homotopy theory in the coarse sense is equivalent to homotopy theory carried out over the base space BC2𝐵subscript𝐶2BC_{2}. We refer to [shulman_parametrized_2008, Sec. 8] for a sophisticated general account of this equivalence. Specifically, the Borel construction XX×C2EC2maps-to𝑋superscriptsubscript𝐶2𝑋𝐸subscript𝐶2X\mapsto X\times^{C_{2}}EC_{2} and the relative mapping space YMapBC2(EC2,Y)maps-to𝑌subscriptMap𝐵subscript𝐶2𝐸subscript𝐶2𝑌Y\mapsto\operatorname{Map}_{BC_{2}}(EC_{2},Y) form a Quillen equivalence between C2subscript𝐶2C_{2}-equivariant spaces with the coarse structure, and spaces over BC2𝐵subscript𝐶2BC_{2}, endowed with what [shulman_parametrized_2008] calls the “mixed” structure on spaces over BC2𝐵subscript𝐶2BC_{2}.

Proposition 8.1.4.

Suppose X𝑋X and Y𝑌Y are C2subscript𝐶2C_{2}-spaces with X𝑋X being a C2subscript𝐶2C_{2}-CW-complex. Then the Borel construction ()×C2EC2superscriptsubscript𝐶2𝐸subscript𝐶2(\cdot)\times^{C_{2}}EC_{2} gives rise to a natural bijection

{coarse C2-homotopy classesof maps XY}{homotopy classes of mapsX×C2EC2Y×C2EC2 over BC2}.coarse C2-homotopy classesof maps XYhomotopy classes of mapsX×C2EC2Y×C2EC2 over BC2\left\{\begin{array}[]{c}\text{coarse $C_{2}$-homotopy classes}\\ \text{of maps $X\to Y$}\end{array}\right\}~{}\cong~{}\left\{\begin{array}[]{c}\text{homotopy classes of maps}\\ \text{$X\times^{C_{2}}EC_{2}\to Y\times^{C_{2}}EC_{2}$ over $BC_{2}$}\end{array}\right\}\ .

8.2. Equivariant Bundles and Classifying Spaces

There is a general theory of equivariant bundles and classifying spaces, more general indeed than what is required in this paper. All examples we consider are of the following form:

Definition 8.2.1.

Suppose we are given a topological C2subscript𝐶2C_{2}-group G𝐺G, or equivalently, a topological group G𝐺G equipped with an involutary automorphism τ:GG:𝜏𝐺𝐺\tau:G\to G. A principal G𝐺G-bundle with a τ𝜏\tau-involution, or just principal G𝐺G-bundle with involution, on a C2subscript𝐶2C_{2}-space X𝑋X is a map π:EX:𝜋𝐸𝑋\pi:E\to X in C2subscript𝐶2C_{2}-spaces such that:

  1. (1)

    π:EX:𝜋𝐸𝑋\pi:E\to X is a principal G𝐺G-bundle,

  2. (2)

    the actions of C2subscript𝐶2C_{2} and of G𝐺G on E𝐸E are compatible, in the sense that if cC2𝑐subscript𝐶2c\in C_{2}, gG𝑔𝐺g\in G and eE𝑒𝐸e\in E, then

    c(eg)=(ce)(cg).𝑐𝑒𝑔𝑐𝑒𝑐𝑔c\cdot(e\cdot g)=(c\cdot e)\cdot(c\cdot g).
Remark 8.2.2.

This concept admits an equivalent definition. Any G𝐺G-bundle E𝐸E, equivariant or not, may be pulled back along the involution λ𝜆\lambda of X𝑋X, in order to form λEXsuperscript𝜆𝐸𝑋\lambda^{*}E\to X. One may then twist the the G𝐺G-action on λEsuperscript𝜆𝐸\lambda^{*}E by changing the structure group along τ:GG:𝜏𝐺𝐺\tau:G\to G, forming E:=G×τλEassignsuperscript𝐸subscript𝜏𝐺superscript𝜆𝐸E^{*}:=G\times_{\tau}\lambda^{*}E. This may be identified with λEEsuperscript𝜆𝐸𝐸\lambda^{*}E\cong E as a topological space over X𝑋X, but with a different G𝐺G-action. The definition of principal G𝐺G-bundle with involution given above is equivalent to asking that π:EX:𝜋𝐸𝑋\pi:E\to X be a principal G𝐺G-bundle together with a G𝐺G-bundles morphism f𝑓f of order 222 from π:EX:𝜋𝐸𝑋\pi:E\to X to π:EX:𝜋superscript𝐸𝑋\pi:E^{*}\to X. On the underlying spaces, f𝑓f must be an isomorphism of order 222 of E𝐸E over X𝑋X, which is equivalent to a C2subscript𝐶2C_{2}-action on E𝐸E making π:EX:𝜋𝐸𝑋\pi:E\to X equivariant. The fact that f:EE:𝑓𝐸superscript𝐸f:E\to E^{*} is an isomorphism of principal G𝐺G-bundles is exactly the relation c(eg)=(ce)(cg)𝑐𝑒𝑔𝑐𝑒𝑐𝑔c\cdot(e\cdot g)=(c\cdot e)\cdot(c\cdot g) above.

Because the automorphism τ:GG:𝜏𝐺𝐺\tau:G\to G is not assumed to be trivial, this notion is more general than the most basic notion of ‘equivariant principal G𝐺G-bundle’, but at the same time, because the sequence

1GGC2C211𝐺right-normal-factor-semidirect-product𝐺subscript𝐶2subscript𝐶211\to G\to G\rtimes C_{2}\to C_{2}\to 1

is split, it is less general than the most general case considered in [may_remarks_1990].

One may construct a C2subscript𝐶2C_{2}-equivariant classifying space for C2subscript𝐶2C_{2}-equivariant principal G𝐺G-bundles, as in [may_remarks_1990]*Thm. 5. We will take the time to explain the procedure, since some of the details will be important later 111We remark that in our case, the group called ΓΓ\Gamma in [may_remarks_1990] is a semidirect product, so EG×EC2𝐸𝐺𝐸subscript𝐶2EG\times EC_{2}, with an appropriate ΓΓ\Gamma-action, is a model for EΓ𝐸ΓE\Gamma. This allows us to replace the space of sections of EC2EΓ𝐸subscript𝐶2𝐸ΓEC_{2}\to E\Gamma by the space of maps EC2EG𝐸subscript𝐶2𝐸𝐺EC_{2}\to EG, an argument that appears in [Guillou2017]*Sec. 5, p. 21.

Notation 8.2.3.

The notation EGBG𝐸𝐺𝐵𝐺EG\to BG will be used for a construction of the classifying space of a topological group G𝐺G, functorial in G𝐺G.

By functoriality, if G𝐺G admits a C2subscript𝐶2C_{2}-action, then EGBG𝐸𝐺𝐵𝐺EG\to BG admits a C2subscript𝐶2C_{2}-action. While the ordinary homotopy type of EGBG𝐸𝐺𝐵𝐺EG\to BG is well defined, irrespective of the model we choose, the C2subscript𝐶2C_{2}-equivariant type is not. The construction EC2GBC2Gsubscript𝐸subscript𝐶2𝐺subscript𝐵subscript𝐶2𝐺E_{C_{2}}G\to B_{C_{2}}G outlined below is a specific choice of such a type.

Start with a EGBG𝐸𝐺𝐵𝐺EG\to BG. Now consider the space of continuous functions 𝒞(EC2,EG)𝒞𝐸subscript𝐶2𝐸𝐺\mathcal{C}(EC_{2},EG). It is endowed with both a G𝐺G-action, induced directly by the G𝐺G-action on EG𝐸𝐺EG, and by a C2subscript𝐶2C_{2}-action given by conjugation of the map. The two actions together induce an action of GC2right-normal-factor-semidirect-product𝐺subscript𝐶2{G\rtimes C_{2}} on 𝒞(EC2,EG)𝒞𝐸subscript𝐶2𝐸𝐺\mathcal{C}(EC_{2},EG), which is contractible, and consequently a C2subscript𝐶2C_{2}-action on 𝒞(EC2,EG)/G𝒞𝐸subscript𝐶2𝐸𝐺𝐺\mathcal{C}(EC_{2},EG)/G, which is a model for BG𝐵𝐺BG. The resulting map

𝒞(EC2,EG)𝒞(EC2,EG)/G𝒞𝐸subscript𝐶2𝐸𝐺𝒞𝐸subscript𝐶2𝐸𝐺𝐺\mathcal{C}(EC_{2},EG)\to\mathcal{C}(EC_{2},EG)/G

is a map of C2subscript𝐶2C_{2}-spaces, and will be denoted

EC2GBC2G.subscript𝐸subscript𝐶2𝐺subscript𝐵subscript𝐶2𝐺E_{C_{2}}G\to B_{C_{2}}G.

We remark that in [may_remarks_1990] and other sources, May and coauthors denote these spaces E(G;GC2)𝐸𝐺right-normal-factor-semidirect-product𝐺subscript𝐶2E(G;G\rtimes C_{2}) and B(G;GC2)𝐵𝐺right-normal-factor-semidirect-product𝐺subscript𝐶2B(G;G\rtimes C_{2}).

Furthermore, the map EC2𝐸subscript𝐶2EC_{2}\to\ast induces a map EG=𝒞(,EG)𝒞(EC2,EG)=EC2G𝐸𝐺𝒞𝐸𝐺𝒞𝐸subscript𝐶2𝐸𝐺subscript𝐸subscript𝐶2𝐺EG=\mathcal{C}(\ast,EG)\to\mathcal{C}(EC_{2},EG)=E_{C_{2}}G. This map is GC2right-normal-factor-semidirect-product𝐺subscript𝐶2G\rtimes C_{2}-equivariant, and induces a C2subscript𝐶2C_{2}-equivariant commutative square

(17) EG𝐸𝐺\textstyle{EG\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}similar-to\scriptstyle{\sim}EC2Gsubscript𝐸subscript𝐶2𝐺\textstyle{E_{C_{2}}G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}BG𝐵𝐺\textstyle{BG\ignorespaces\ignorespaces\ignorespaces\ignorespaces}similar-to\scriptstyle{\sim}BC2G,subscript𝐵subscript𝐶2𝐺\textstyle{B_{C_{2}}G,}

in which the horizontal maps are coarse, but not necessarily fine, C2subscript𝐶2C_{2}-weak equivalences. The map EC2GBC2Gsubscript𝐸subscript𝐶2𝐺subscript𝐵subscript𝐶2𝐺E_{C_{2}}G\to B_{C_{2}}G is a classifying space for principal G𝐺G-bundles with involution.

Proposition 8.2.4.

If X𝑋X is a C2subscript𝐶2C_{2}-CW-complex, then there is a natural bijection between [X,BC2G]C2subscript𝑋subscript𝐵subscript𝐶2𝐺subscript𝐶2[X,B_{C_{2}}G]_{C_{2}} and the set of isomorphism classes of principal G𝐺G-bundles with involution on X𝑋X.

We refer to [may_remarks_1990, Thm. 5] for the proof.

Proposition 8.2.5.

If X𝑋X is a free C2subscript𝐶2C_{2}-CW-complex, then the following are naturally isomorphic

  1. (a)

    [X,BC2G]C2subscript𝑋subscript𝐵subscript𝐶2𝐺subscript𝐶2[X,B_{C_{2}}G]_{C_{2}},

  2. (b)

    [X,BC2G]C2-coarsesubscript𝑋subscript𝐵subscript𝐶2𝐺subscript𝐶2-coarse[X,B_{C_{2}}G]_{C_{2}\text{-coarse}},

  3. (c)

    [X,BG]C2subscript𝑋𝐵𝐺subscript𝐶2[X,BG]_{C_{2}},

  4. (d)

    [X,BG]C2-coarsesubscript𝑋𝐵𝐺subscript𝐶2-coarse[X,BG]_{C_{2}\text{-coarse}},

  5. (e)

    The set of isomorphism classes of principal G𝐺G bundles with involution on X𝑋X.

Proof.

The equivalences all follow from Propositions 8.1.3, 8.2.4 and Diagram (17). ∎

Remark 8.2.6.

Proposition 8.2.5 means that if one is willing to restrict one’s attention to spaces with free C2subscript𝐶2C_{2}-action, then the construction of EC2GBC2Gsubscript𝐸subscript𝐶2𝐺subscript𝐵subscript𝐶2𝐺E_{C_{2}}G\to B_{C_{2}}G from EGBG𝐸𝐺𝐵𝐺EG\to BG is not necessary. The C2subscript𝐶2C_{2}-action given by the functoriality of the construction of BG𝐵𝐺BG is sufficient.

Remark 8.2.7.

Let G𝐺G be a topological group. One may give G×G𝐺𝐺G\times G the C2subscript𝐶2C_{2}-action which interchanges the two factors. Then the resulting classifying space BG×BG𝐵𝐺𝐵𝐺BG\times BG also admits this action. In this instance, the space BC2(G×G)subscript𝐵subscript𝐶2𝐺𝐺B_{C_{2}}(G\times G) is C2subscript𝐶2C_{2}-equivalent to BG×BG𝐵𝐺𝐵𝐺BG\times BG with the interchange action, which may be verified by testing on C2subscript𝐶2C_{2}-fixed points, for instance.

The construction of taking a space Y𝑌Y and producing Y×Y𝑌𝑌Y\times Y with the C2subscript𝐶2C_{2}-action interchanging the factors is right adjoint to the forgetful functor. Suppose X𝑋X is a C2subscript𝐶2C_{2}-space, then

(18) [X,BG][X,BG×BG]C2𝑋𝐵𝐺subscript𝑋𝐵𝐺𝐵𝐺subscript𝐶2[X,BG]\cong[X,BG\times BG]_{C_{2}}

where the set on the left is the set of maps in the nonequivariant homotopy category.

8.3. The Case of PGLnsubscriptPGL𝑛\operatorname{PGL}_{n}-bundles

For the rest of this section, we write GLnsubscriptGL𝑛\operatorname{GL}_{n}, PGLnsubscriptPGL𝑛\operatorname{PGL}_{n} etc. for the Lie group of complex points, GLn()subscriptGL𝑛\operatorname{GL}_{n}(\mathbb{C}), PGLn()subscriptPGL𝑛\operatorname{PGL}_{n}(\mathbb{C}) and so on.

We now specify C2subscript𝐶2C_{2}-actions on groups that will appear in the sequel. There is a C2subscript𝐶2C_{2}-action on GLnsubscriptGL𝑛\operatorname{GL}_{n} in which the non-trivial element acts via AAtrmaps-to𝐴superscript𝐴trA\mapsto A^{-\text{\rm tr}}, the transpose-inverse. This passes to certain subquotients of GLnsubscriptGL𝑛\operatorname{GL}_{n}, and we will use it as the C2subscript𝐶2C_{2}-action on the groups μnsubscript𝜇𝑛\mu_{n}, ×superscript\mathbb{C}^{\times}, SLnsubscriptSL𝑛\operatorname{SL}_{n} and PGLnsubscriptPGL𝑛\operatorname{PGL}_{n}, all viewed either as subgroups or as quotients of GLnsubscriptGL𝑛\operatorname{GL}_{n}. Specifically, we write tr:PGLnPGLn:trsubscriptPGL𝑛subscriptPGL𝑛-\text{\rm tr}:\operatorname{PGL}_{n}\to\operatorname{PGL}_{n} for the outer automorphism AAtrmaps-to𝐴superscript𝐴trA\mapsto A^{-\text{\rm tr}}.

There is also a C2subscript𝐶2C_{2}-action on GLn×GLnsubscriptGL𝑛subscriptGL𝑛\operatorname{GL}_{n}\times\operatorname{GL}_{n} given by interchanging the factors and then applying the transpose-inverse, so that the induced involution is

(A,B)(Btr,Atr).maps-to𝐴𝐵superscript𝐵trsuperscript𝐴tr(A,B)\mapsto(B^{-\text{\rm tr}},A^{-\text{\rm tr}})\ .

This will be used for certain subquotients of this group, including μn×μnsubscript𝜇𝑛subscript𝜇𝑛\mu_{n}\times\mu_{n}, ×××superscriptsuperscript\mathbb{C}^{\times}\times\mathbb{C}^{\times}, SLn×SLnsubscriptSL𝑛subscriptSL𝑛\operatorname{SL}_{n}\times\operatorname{SL}_{n} and PGLn×PGLnsubscriptPGL𝑛subscriptPGL𝑛\operatorname{PGL}_{n}\times\operatorname{PGL}_{n}.

There is a diagonal inclusion GLnGLn×GLnsubscriptGL𝑛subscriptGL𝑛subscriptGL𝑛\operatorname{GL}_{n}\to\operatorname{GL}_{n}\times\operatorname{GL}_{n}, given by A(A,A)maps-to𝐴𝐴𝐴A\mapsto(A,A). It is C2subscript𝐶2C_{2}-equivariant, and induces similar maps for the aforementioned subquotients of GLnsubscriptGL𝑛\operatorname{GL}_{n}.

One may form C2subscript𝐶2C_{2}-equivariant classifying spaces for the groups named above, as outlined in Subsection 8.2. Among the possibilities, two are particularly useful to us: BC2PGLnsubscript𝐵subscript𝐶2subscriptPGL𝑛B_{C_{2}}\operatorname{PGL}_{n} and BC2(PGLn×PGLn)subscript𝐵subscript𝐶2subscriptPGL𝑛subscriptPGL𝑛B_{C_{2}}(\operatorname{PGL}_{n}\times\operatorname{PGL}_{n}).

Proposition 8.3.1.

Let X𝑋X be a C2subscript𝐶2C_{2}-CW-complex with corresponding involution λ𝜆\lambda, and let n𝑛n be a natural number. Then the following sets are in natural bijective correspondence:

  1. (a)

    Isomorphism classes of degree-n𝑛n topological Azumaya algebras with λ𝜆\lambda-involution on X𝑋X,

  2. (b)

    Isomorphism classes of principal PGLnsubscriptPGL𝑛\operatorname{PGL}_{n}-bundles with involution on X𝑋X,

  3. (c)

    [X,BC2PGLn]C2subscript𝑋subscript𝐵subscript𝐶2subscriptPGL𝑛subscript𝐶2[X,B_{C_{2}}\operatorname{PGL}_{n}]_{C_{2}}.

Proof.

There is a well-known bijection between Azumaya algebras of degree n𝑛n on X𝑋X and principal PGLnsubscriptPGL𝑛\operatorname{PGL}_{n}-bundles, since PGLn()subscriptPGL𝑛\operatorname{PGL}_{n}(\mathbb{C}) is the automorphism group of M:=Matn×n()assign𝑀subscriptMat𝑛𝑛M:=\operatorname{Mat}_{n\times n}(\mathbb{C}) as a \mathbb{C}-algebra, see Subsection 2.5. Let A𝐴A be an Azumaya algebra of degree n𝑛n on X𝑋X and P𝑃P the associated principal PGLnsubscriptPGL𝑛\operatorname{PGL}_{n}-bundle.

The functor of taking opposite algebras on Azumaya algebras corresponds to the functor of change of group along tr:PGLnPGLn:trsubscriptPGL𝑛subscriptPGL𝑛-\text{\rm tr}:\operatorname{PGL}_{n}\to\operatorname{PGL}_{n} of principal PGLnsubscriptPGL𝑛\operatorname{PGL}_{n}-bundles; this can be seen at the level of clutching functions. Indeed, note that mmtr:Matn×n()Matn×n()opm\mapsto m^{\text{\rm tr}}:\operatorname{Mat}_{n\times n}(\mathbb{C})\to\operatorname{Mat}_{n\times n}(\mathbb{C})^{\text{op}} is a \mathbb{C}-algebra isomorphism. If one chooses coordinates for A𝐴A on two open sets of X𝑋X on which it trivializes, then the clutching function f:Matn×n()×(U1U2)Matn×n()×(U1U2):𝑓subscriptMat𝑛𝑛subscript𝑈1subscript𝑈2subscriptMat𝑛𝑛subscript𝑈1subscript𝑈2f:\operatorname{Mat}_{n\times n}(\mathbb{C})\times(U_{1}\cap U_{2})\to\operatorname{Mat}_{n\times n}(\mathbb{C})\times(U_{1}\cap U_{2}) given by mxmx1maps-to𝑚𝑥𝑚superscript𝑥1m\mapsto xmx^{-1}, for some x:(U1U2)PGLn():𝑥subscript𝑈1subscript𝑈2subscriptPGL𝑛x:(U_{1}\cap U_{2})\to\operatorname{PGL}_{n}(\mathbb{C}). For the same choice of coordinates over both U1subscript𝑈1U_{1} and U2subscript𝑈2U_{2}, the clutching function fopsuperscript𝑓opf^{\text{op}} of the opposite algebra is given by mtr(xmx1)tr=xtrmtrxtrmaps-tosuperscript𝑚trsuperscript𝑥𝑚superscript𝑥1trsuperscript𝑥trsuperscript𝑚trsuperscript𝑥trm^{\text{\rm tr}}\mapsto(xmx^{-1})^{\text{\rm tr}}=x^{-\text{\rm tr}}m^{\text{\rm tr}}x^{\text{\rm tr}}.

Therefore, the data of an isomorphism of AAop𝐴superscript𝐴opA\to A^{\text{op}} of order 222 over the involution λ:XX:𝜆𝑋𝑋\lambda:X\to X is equivalent to an order-222 self-map of the associated principal PGLnsubscriptPGL𝑛\operatorname{PGL}_{n}-bundle, PP𝑃superscript𝑃P\to P^{*} over X𝑋X, where Psuperscript𝑃P^{*} denotes the principal PGLnsubscriptPGL𝑛\operatorname{PGL}_{n}-bundle

P:=PGLn×trλP.assignsuperscript𝑃subscripttrsubscriptPGL𝑛superscript𝜆𝑃P^{*}:=\operatorname{PGL}_{n}\times_{-\text{\rm tr}}\lambda^{*}P.

As explained in Remark 8.2.2, this is equivalent to the definition of principal PGLnsubscriptPGL𝑛\operatorname{PGL}_{n}-bundle with involution in Definition 8.2.1; thus establishing the equivalence of (a) and (b).

The equivalence between (b) and (c) is an application of Proposition 8.2.4. ∎

The space BC2(PGLn×PGLn)subscript𝐵subscript𝐶2subscriptPGL𝑛subscriptPGL𝑛B_{C_{2}}(\operatorname{PGL}_{n}\times\operatorname{PGL}_{n}), by similar methods, is seen to classify ordered pairs of PGLnsubscriptPGL𝑛\operatorname{PGL}_{n}-bundles on a C2subscript𝐶2C_{2}-space X𝑋X, such that the one is obtained from the other by twisting relative to the involutions of X𝑋X and PGLnsubscriptPGL𝑛\operatorname{PGL}_{n}. But the category of such ordered pairs is identical to the category of ordinary PGLnsubscriptPGL𝑛\operatorname{PGL}_{n}-bundles on the space X𝑋X, forgetting the C2subscript𝐶2C_{2}-action.

This last fact also manifests itself algebraically via Remark 8.2.7 in the following way: Suppose G𝐺G is a subgroup of GLnsubscriptGL𝑛\operatorname{GL}_{n} closed under taking transposes, or a quotient of GLnsubscriptGL𝑛\operatorname{GL}_{n} by such a subgroup, let (G×G,α)𝐺𝐺𝛼(G\times G,\alpha) denote the product group with the involution (A,B)(Btr,Atr)maps-to𝐴𝐵superscript𝐵trsuperscript𝐴tr(A,B)\mapsto(B^{-\text{\rm tr}},A^{-\text{\rm tr}}), and let (G×G,i)𝐺𝐺𝑖(G\times G,i) denote the product group with the involution exchanging A𝐴A and B𝐵B. Then (A,B)(A,Btr)maps-to𝐴𝐵𝐴superscript𝐵tr(A,B)\mapsto(A,B^{-\text{\rm tr}}) is a C2subscript𝐶2C_{2}-equivariant isomorphism between these two groups with involution.

We will apply the classifying space theory developed above in the two extreme cases where the C2subscript𝐶2C_{2}-action on X𝑋X is trivial and when it is free.

8.4. Trivial Action

Suppose X𝑋X is equipped with a trivial C2subscript𝐶2C_{2}-action. Then principal PGLnsubscriptPGL𝑛\operatorname{PGL}_{n}-bundles with involution on X𝑋X are classified by [X,BC2PGLn]C2=[X,(BC2PGLn)C2]subscript𝑋subscript𝐵subscript𝐶2subscriptPGL𝑛subscript𝐶2𝑋superscriptsubscript𝐵subscript𝐶2subscriptPGL𝑛subscript𝐶2[X,B_{C_{2}}\operatorname{PGL}_{n}]_{C_{2}}=[X,(B_{C_{2}}\operatorname{PGL}_{n})^{C_{2}}].

Proposition 8.4.1.

Let n𝑛n be a positive integer. Then the fixed locus (BC2PGLn)C2superscriptsubscript𝐵subscript𝐶2subscriptPGL𝑛subscript𝐶2(B_{C_{2}}\operatorname{PGL}_{n})^{C_{2}} is homeomorphic to

  1. (i)

    BPOnBPSpnsquare-union𝐵subscriptPO𝑛𝐵subscriptPSp𝑛B\operatorname{PO}_{n}\,\sqcup\,B\operatorname{PSp}_{n} if n𝑛n is even;

  2. (ii)

    BPOn𝐵subscriptPO𝑛B\operatorname{PO}_{n} if n𝑛n is odd.

Proof.

We may calculate the fixed-point-sets of BC2(PGLn)subscript𝐵subscript𝐶2subscriptPGL𝑛B_{C_{2}}(\operatorname{PGL}_{n}) by means of [may_remarks_1990]*Thm. 7. We explain the application of this theorem in the current case.

If APGLn𝐴subscriptPGL𝑛A\in\operatorname{PGL}_{n} is a matrix such that AAtr=In𝐴superscript𝐴trsubscript𝐼𝑛AA^{-\text{\rm tr}}=I_{n}, then (A,tr)Γ:=PGLnC2𝐴trΓassignright-normal-factor-semidirect-productsubscriptPGL𝑛subscript𝐶2(A,-\text{\rm tr})\in\Gamma:=\operatorname{PGL}_{n}\rtimes C_{2} generates a subgroup that maps isomorphically onto C2subscript𝐶2C_{2} and intersects PGLnsubscriptPGL𝑛\operatorname{PGL}_{n} trivially. Denote by PGLn(A,tr)superscriptsubscriptPGL𝑛𝐴tr\operatorname{PGL}_{n}^{(A,-\text{\rm tr})} the commutant of (A,tr)𝐴tr(A,-\text{\rm tr}) in PGLnsubscriptPGL𝑛\operatorname{PGL}_{n}, i.e., the subgroup of PGLnsubscriptPGL𝑛\operatorname{PGL}_{n} consisting of elements X𝑋X such that Xtr=A1XAsuperscript𝑋trsuperscript𝐴1𝑋𝐴X^{-\text{\rm tr}}=A^{-1}XA. We write AAsimilar-to𝐴superscript𝐴A\sim A^{\prime} if (A,tr)𝐴tr(A,-\text{\rm tr}) and (A,tr)superscript𝐴tr(A^{\prime},-\text{\rm tr}) are conjugate under PGLnsubscriptPGL𝑛\operatorname{PGL}_{n}, or equivalently, if there exists XPGLn𝑋subscriptPGL𝑛X\in\operatorname{PGL}_{n} such that XAXtr=A𝑋𝐴superscript𝑋trsuperscript𝐴XAX^{\text{\rm tr}}=A^{\prime}. Then the theorem asserts that

(BC2PGLn)C2=AB(PGL(A,tr))superscriptsubscript𝐵subscript𝐶2subscriptPGL𝑛subscript𝐶2subscriptsquare-union𝐴𝐵superscriptPGL𝐴tr(B_{C_{2}}\operatorname{PGL}_{n})^{C_{2}}=\bigsqcup_{A}B(\operatorname{PGL}^{(A,-\text{\rm tr})})

as A𝐴A runs over equivalence classes of elements APGLn𝐴subscriptPGL𝑛A\in\operatorname{PGL}_{n} satisfying AtrA=Insuperscript𝐴tr𝐴subscript𝐼𝑛A^{-\text{\rm tr}}A=I_{n}.

When n𝑛n is even, say n=2m𝑛2𝑚n=2m, there are two such equivalence classes, namely the class of Insubscript𝐼𝑛I_{n} and the class of h2m(1)subscript2𝑚1h_{2m}(-1), in the notation of Example 7.1.1, as can be calculated directly. The fixed points under the action are those matrices for which Btr=Bsuperscript𝐵tr𝐵B^{-\text{\rm tr}}=B in the first case and Btr=h2m(1)Bh2m(1)1superscript𝐵trsubscript2𝑚1𝐵subscript2𝑚superscript11B^{-\text{\rm tr}}=h_{2m}(-1)Bh_{2m}(-1)^{-1} in the second, which is to say, the subgroups of orthogonal and of symplectic matrices respectively. We therefore deduce

(BC2PGLn)C2=BPOnBPSpn.superscriptsubscript𝐵subscript𝐶2subscriptPGL𝑛subscript𝐶2square-union𝐵subscriptPO𝑛𝐵subscriptPSp𝑛(B_{C_{2}}\operatorname{PGL}_{n})^{C_{2}}=B\operatorname{PO}_{n}\,\sqcup\,B\operatorname{PSp}_{n}.

When n𝑛n is odd, the argument is much the same, but only BPOn𝐵subscriptPO𝑛B\operatorname{PO}_{n} occurs. ∎

Remark 8.4.2.

By Theorem 5.4.5, we know that there are two types of involutions on Azumaya algebras over connected topological spaces with trivial action, the symplectic and orthogonal. By means of Examples 7.1.2 and 7.1.4, we know that the orthogonal and symplectic Azumaya algebras with involution are equivalent to principal bundles for the groups POnsubscriptPO𝑛\operatorname{PO}_{n} and PSpnsubscriptPSp𝑛\operatorname{PSp}_{n}, the latter when n𝑛n is even. Proposition 8.4.1 has recovered these observations via equivariant homotopy theory.

8.5. Free Action

Now we address the case where the action of C2subscript𝐶2C_{2} on X𝑋X is free. In this case, the quotient map XY:=X/C2𝑋𝑌assign𝑋subscript𝐶2X\to Y:=X/C_{2} is a two-sheeted covering space map.

Proposition 8.5.1.

Let X𝑋X be a free C2subscript𝐶2C_{2}-CW-complex, with C2subscript𝐶2C_{2} acting by the involution λ𝜆\lambda, and let Y=X/C2𝑌𝑋subscript𝐶2Y=X/C_{2}. Consider Y𝑌Y as a space over BC2𝐵subscript𝐶2BC_{2}, or alternatively, as a space equipped with a distinguished class αH1(Y,C2)𝛼superscriptH1𝑌subscript𝐶2\alpha\in\mathrm{H}^{1}(Y,C_{2}). There are natural bijections between the following:

  1. (a)

    Isomorphism classes of degree-n𝑛n topological Azumaya algebras over X𝑋X equipped with a λ𝜆\lambda-involution.

  2. (b)

    [X,BPGLn]C2subscript𝑋𝐵subscriptPGL𝑛subscript𝐶2[X,B\operatorname{PGL}_{n}]_{C_{2}}.

  3. (c)

    [Y,B(PGLnC2)]BC2subscript𝑌𝐵right-normal-factor-semidirect-productsubscriptPGL𝑛subscript𝐶2𝐵subscript𝐶2[Y,B(\operatorname{PGL}_{n}\rtimes C_{2})]_{BC_{2}}.

  4. (d)

    Elements of the preimage of α𝛼\alpha under H1(Y,PGLnC2)H1(Y,C2)superscriptH1𝑌right-normal-factor-semidirect-productsubscriptPGL𝑛subscript𝐶2superscriptH1𝑌subscript𝐶2\mathrm{H}^{1}(Y,\operatorname{PGL}_{n}\rtimes C_{2})\to\mathrm{H}^{1}(Y,C_{2}).

Proof.

Propositions 8.3.1 and 8.2.5 give the bijection between (a) and (b), and Proposition 8.1.4 gives a bijection between (b) and (c). The one-to-one correspondence between (c) and (d) is standard. ∎

We continue to assume that X𝑋X is a free C2subscript𝐶2C_{2}-CW-complex and let Y=X/C2𝑌𝑋subscript𝐶2Y=X/C_{2}. We would like to have a classifying-space-level understanding of the cohomological transfer map transfX/Y:H2(X,𝔾m)H2(Y,𝔾m):subscripttransf𝑋𝑌superscriptH2𝑋subscript𝔾𝑚superscriptH2𝑌subscript𝔾𝑚\operatorname{transf}_{X/Y}:\mathrm{H}^{2}(X,\mathbb{G}_{m})\to\mathrm{H}^{2}(Y,\mathbb{G}_{m}) considered in Subsection 6.2.

To that end, let μ𝜇\mu denote the discrete group μnsubscript𝜇𝑛\mu_{n} or the topological group ×superscript{\mathbb{C}^{\times}}. We endow μ𝜇\mu with the involution aa1maps-to𝑎superscript𝑎1a\mapsto a^{-1}, give μ×μ𝜇𝜇\mu\times\mu the involution (a,b)(b1,a1)maps-to𝑎𝑏superscript𝑏1superscript𝑎1(a,b)\mapsto(b^{-1},a^{-1}), and let μtrivsuperscript𝜇triv\mu^{\text{triv}} denote μ𝜇\mu with the trivial action.

The map μ×μμtriv𝜇𝜇superscript𝜇triv\mu\times\mu\to\mu^{\text{triv}} defined by (a,b)ab1maps-to𝑎𝑏𝑎superscript𝑏1(a,b)\mapsto ab^{-1} is C2subscript𝐶2C_{2}-equivariant, and its kernel consists of pairs of the form (a,a)𝑎𝑎(a,a), which is the image of the diagonal map μμ×μ𝜇𝜇𝜇\mu\to\mu\times\mu. That is, there is a C2subscript𝐶2C_{2}-equivariant short exact sequence of C2subscript𝐶2C_{2}-groups

1μμ×μμtriv11𝜇𝜇𝜇superscript𝜇triv11\to\mu\to\mu\times\mu\to\mu^{\text{triv}}\to 1

and therefore, a sequence of C2subscript𝐶2C_{2}-equivariant maps in which any three consecutive terms form a homotopy fibre sequence:

μμ×μμtrivBμB(μ×μ)BμtrivB2μ.𝜇𝜇𝜇superscript𝜇triv𝐵𝜇𝐵𝜇𝜇𝐵superscript𝜇trivsuperscript𝐵2𝜇\mu\to\mu\times\mu\to\mu^{\text{triv}}\to B\mu\to B(\mu\times\mu)\to B\mu^{\text{triv}}\to B^{2}\mu\to\cdots.

Any such homotopy fibre sequence is a homotopy fibre sequence in the C2subscript𝐶2C_{2}-equivariant coarse structure. These constructions are plainly natural with respect to inclusion of subgroups of ×superscript\mathbb{C}^{\times}.

Now, if X𝑋X is a free C2subscript𝐶2C_{2}-CW-complex, then thanks to Proposition 8.2.5, one arrives at a long exact sequence of abelian groups

(19) [X,Biμ]C2[X,Biμ×Biμ]C2[X,Biμtriv]C2.subscript𝑋superscript𝐵𝑖𝜇subscript𝐶2subscript𝑋superscript𝐵𝑖𝜇superscript𝐵𝑖𝜇subscript𝐶2subscript𝑋superscript𝐵𝑖superscript𝜇trivsubscript𝐶2\dots\to[X,B^{i}\mu]_{C_{2}}\to[X,B^{i}\mu\times B^{i}\mu]_{C_{2}}\to[X,B^{i}\mu^{\text{triv}}]_{C_{2}}\to\cdots.

Since μ×μ𝜇𝜇\mu\times\mu with this action is isomorphic to μ×μ𝜇𝜇\mu\times\mu with the interchange action, it follows that [X,Biμ×Biμ]C2[X,Biμ]=Hi(X,μ)subscript𝑋superscript𝐵𝑖𝜇superscript𝐵𝑖𝜇subscript𝐶2𝑋superscript𝐵𝑖𝜇superscriptH𝑖𝑋𝜇[X,B^{i}\mu\times B^{i}\mu]_{C_{2}}\cong[X,B^{i}\mu]=\mathrm{H}^{i}(X,\mu). Moreover, [X,Biμtriv]C2subscript𝑋superscript𝐵𝑖superscript𝜇trivsubscript𝐶2[X,B^{i}\mu^{\text{triv}}]_{C_{2}} is simply [Y,Biμ]=Hi(Y,μ)𝑌superscript𝐵𝑖𝜇superscriptH𝑖𝑌𝜇[Y,B^{i}\mu]=\mathrm{H}^{i}(Y,\mu).

Therefore, the sequence of (19) reduces in this case to

(20) [X,Biμ]C2Hi(X,μ)transfHi(Y,μ).subscript𝑋superscript𝐵𝑖𝜇subscript𝐶2superscriptH𝑖𝑋𝜇transfsuperscriptH𝑖𝑌𝜇\cdots\to[X,B^{i}\mu]_{C_{2}}\to\mathrm{H}^{i}(X,\mu)\xrightarrow{\operatorname{transf}}\mathrm{H}^{i}(Y,\mu)\to\cdots.

When μ=×𝜇superscript\mu={\mathbb{C}^{\times}} and i=2𝑖2i=2, the map denoted transftransf\operatorname{transf} agrees with the transfer map defined in Subsection 6.2. Indeed, we know that the transfer map in Section 6 agrees with the ordinary transfer map for a 222-sheeted covering in the case at hand, Example 6.2.5. It suffices therefore to show that the map transftransf\operatorname{transf} in (20) is the usual transfer map for a 222-sheeted cover. The trivial case X=Y×C2𝑋𝑌subscript𝐶2X=Y\times C_{2} is elementary. The general case where π:XY:𝜋𝑋𝑌\pi:X\to Y is merely locally trivial can be deduced from the trivial case by viewing Hi(Y,μ)superscriptH𝑖𝑌𝜇\mathrm{H}^{i}(Y,\mu) and Hi(X,μ)superscriptH𝑖𝑋𝜇\mathrm{H}^{i}(X,\mu) as Čech cohomology groups and calculating each using covers 𝒰𝒰\mathcal{U} of Y𝑌Y and π1𝒰superscript𝜋1𝒰\pi^{-1}\mathcal{U} of X𝑋X where 𝒰𝒰\mathcal{U} trivializes the double cover π𝜋\pi.

Proposition 8.5.2.

Let μ𝜇\mu be μnsubscript𝜇𝑛\mu_{n} or ×superscript\mathbb{C}^{\times}, given the involution zz1maps-to𝑧superscript𝑧1z\mapsto z^{-1}. Let X𝑋X be a space with free C2subscript𝐶2C_{2}-action, let Y=X/C2𝑌𝑋subscript𝐶2Y=X/C_{2}, and let ξ:XBiμ:𝜉𝑋superscript𝐵𝑖𝜇\xi:X\to B^{i}\mu be an equivariant map, representing a cohomology class ξHi(X,μ)𝜉superscriptH𝑖𝑋𝜇\xi\in\mathrm{H}^{i}(X,\mu). Then transfX/Yξ=0subscripttransf𝑋𝑌𝜉0\operatorname{transf}_{X/Y}\xi=0.

Proof.

Since ξ:XBnμ:𝜉𝑋superscript𝐵𝑛𝜇\xi:X\to B^{n}\mu is equivariant, and the action on X𝑋X is free, ξ𝜉\xi lies in the image of [X,Bnμ]C2subscript𝑋superscript𝐵𝑛𝜇subscript𝐶2[X,B^{n}\mu]_{C_{2}} in [X,Bnμ]=Hn(X,μ)𝑋superscript𝐵𝑛𝜇superscriptH𝑛𝑋𝜇[X,B^{n}\mu]=\mathrm{H}^{n}(X,\mu). Thus, the result follows from the exact sequence (20). ∎

9. An Azumaya Algebra with no Involution of the Second Kind

We finally construct the example promised at the beginning of Section 8.

Throughout, the notation x𝑥x\mathbb{Z} means a free cyclic group, written additively, with a named generator x𝑥x. Recall that for a topological space X𝑋X, the sheaf cohomology group H2(X,×):=H2(X,𝒞(X,×))assignsuperscriptH2𝑋superscriptsuperscriptH2𝑋𝒞𝑋superscript\mathrm{H}^{2}(X,{\mathbb{C}^{\times}}):=\mathrm{H}^{2}(X,{\mathcal{C}}(X,{\mathbb{C}^{\times}})) is isomorphic to the singular cohomology group H3(X,)superscriptH3𝑋\mathrm{H}^{3}(X,\mathbb{Z}), see [asher_auel_azumaya_2017, §2.1], for instance. We shall use the latter group for the most part.

9.1. A Cohomological Obstruction

In all cases, the groups appearing in this subsection are the complex points of linear algebraic groups. In the interest of brevity, the relevant linear algebraic group, e.g. SLnsubscriptSL𝑛\operatorname{SL}_{n}, will be written in place of the group itself, e.g. SLn()subscriptSL𝑛\operatorname{SL}_{n}(\mathbb{C}).

Unless otherwise specified, groups appearing will be endowed with a C2subscript𝐶2C_{2}-action. For the groups SLnsubscriptSL𝑛\operatorname{SL}_{n}, the action is that sending A𝐴A to Atrsuperscript𝐴trA^{-\text{\rm tr}}, which restricts to the action rr1maps-to𝑟superscript𝑟1r\mapsto r^{-1} on the central subgroup μnsubscript𝜇𝑛\mu_{n}. For the groups SLn×SLnsubscriptSL𝑛subscriptSL𝑛\operatorname{SL}_{n}\times\operatorname{SL}_{n}, the action is that given by (A,B)(Btr,Atr)maps-to𝐴𝐵superscript𝐵trsuperscript𝐴tr(A,B)\mapsto(B^{-\text{\rm tr}},A^{-\text{\rm tr}}), and similarly for μn×μnsubscript𝜇𝑛subscript𝜇𝑛\mu_{n}\times\mu_{n}. The maps SLnSLn×SLnsubscriptSL𝑛subscriptSL𝑛subscriptSL𝑛\operatorname{SL}_{n}\to\operatorname{SL}_{n}\times\operatorname{SL}_{n} and μnμn×μnsubscript𝜇𝑛subscript𝜇𝑛subscript𝜇𝑛\mu_{n}\to\mu_{n}\times\mu_{n} are given by diagonal inclusions.

Embed μnSLn×SLnsubscript𝜇𝑛subscriptSL𝑛subscriptSL𝑛\mu_{n}\hookrightarrow\operatorname{SL}_{n}\times\operatorname{SL}_{n} via r(rIn,rIn)maps-to𝑟𝑟subscript𝐼𝑛𝑟subscript𝐼𝑛r\mapsto(rI_{n},rI_{n}), and let Qnsubscript𝑄𝑛Q_{n} denote the group obtained as the quotient of SLn×SLnsubscriptSL𝑛subscriptSL𝑛\operatorname{SL}_{n}\times\operatorname{SL}_{n} by the image of μnsubscript𝜇𝑛\mu_{n}.

In the following diagram, the horizontal arrows of the first two rows are C2subscript𝐶2C_{2}-equivariant. This induces C2subscript𝐶2C_{2}-actions on the groups in the third row so that all arrows become C2subscript𝐶2C_{2}-equivariant.

Figure 1. A Diagram of C2subscript𝐶2C_{2} Groups
11\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}11\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}11\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces}μnsubscript𝜇𝑛\textstyle{\mu_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}μnsubscript𝜇𝑛\textstyle{\mu_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}μn×μnsubscript𝜇𝑛subscript𝜇𝑛\textstyle{\mu_{n}\times\mu_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}SLnsubscriptSL𝑛\textstyle{\operatorname{SL}_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}SLn×SLnsubscriptSL𝑛subscriptSL𝑛\textstyle{\operatorname{SL}_{n}\times\operatorname{SL}_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}SLn×SLnsubscriptSL𝑛subscriptSL𝑛\textstyle{\operatorname{SL}_{n}\times\operatorname{SL}_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}PGLnsubscriptPGL𝑛\textstyle{\operatorname{PGL}_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Qnsubscript𝑄𝑛\textstyle{Q_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}PGLn×PGLnsubscriptPGL𝑛subscriptPGL𝑛\textstyle{\operatorname{PGL}_{n}\times\operatorname{PGL}_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}11\textstyle{1}11\textstyle{1}1.1\textstyle{1.}

Each of the groups appearing above is equipped with a C2subscript𝐶2C_{2}-action, and consequently each may be extended to a semidirect product with C2subscript𝐶2C_{2}, and equivariant classifying spaces of the form BC2Gsubscript𝐵subscript𝐶2𝐺B_{C_{2}}G may be constructed as in Subsection 8.2. Since we will consider equivariant maps with free C2subscript𝐶2C_{2}-action on the source, by Proposition 8.2.5, we may use any functorial model of BG𝐵𝐺BG with its functorially-induced C2subscript𝐶2C_{2}-action instead.

Proposition 9.1.1.

The C2subscript𝐶2C_{2}-action on PGLnsubscriptPGL𝑛\operatorname{PGL}_{n} induces an action on H(BPGLn,)superscriptH𝐵subscriptPGL𝑛\mathrm{H}^{*}(B\operatorname{PGL}_{n},\mathbb{Z}). In low degrees, this action is summarized by Table 1.

i𝑖i Hi(BPGLn,)superscriptH𝑖𝐵subscriptPGL𝑛\mathrm{H}^{i}(B\operatorname{PGL}_{n},\mathbb{Z}) Action
0 \mathbb{Z} trivial
1 0 -
2 0 -
3 α/n𝛼𝑛\alpha\mathbb{Z}/n ααmaps-to𝛼𝛼\alpha\mapsto-\alpha
4 c~2subscript~𝑐2\tilde{c}_{2}\mathbb{Z} trivial
Table 1. The C2subscript𝐶2C_{2}-action on H(BPGLn,)superscriptH𝐵subscriptPGL𝑛\mathrm{H}^{*}(B\operatorname{PGL}_{n},\mathbb{Z}).
Proof.

The compatible C2subscript𝐶2C_{2}-actions on the terms of the exact sequence 1μnSLnPGLn11subscript𝜇𝑛subscriptSL𝑛subscriptPGL𝑛11\to\mu_{n}\to\operatorname{SL}_{n}\to\operatorname{PGL}_{n}\to 1 induce an action on the fibre sequence BSLnBPGLnB2μn𝐵subscriptSL𝑛𝐵subscriptPGL𝑛superscript𝐵2subscript𝜇𝑛B\operatorname{SL}_{n}\to B\operatorname{PGL}_{n}\to B^{2}\mu_{n}, and therefore an action of C2subscript𝐶2C_{2} on the associated Serre spectral sequence, which is illustrated in Figure 2.

c2subscript𝑐2\textstyle{\mathbb{Z}c_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}d5subscript𝑑5\scriptstyle{d_{5}}00\textstyle{0}00\textstyle{0}\textstyle{\mathbb{Z}}00\textstyle{0}00\textstyle{0}α/n𝛼𝑛\textstyle{\alpha\mathbb{Z}/n}00\textstyle{0}κ/(ϵn)𝜅italic-ϵ𝑛\textstyle{\kappa\mathbb{Z}/(\epsilon n)}
Figure 2. A portion of the Serre spectral sequence in cohomology associated to BSLnBPGLnB2μn𝐵subscriptSL𝑛𝐵subscriptPGL𝑛superscript𝐵2subscript𝜇𝑛B\operatorname{SL}_{n}\to B\operatorname{PGL}_{n}\to B^{2}\mu_{n}.

Here, ϵitalic-ϵ\epsilon is 222 if n𝑛n is even and is 111 otherwise.

The action on α/n𝛼𝑛\alpha\mathbb{Z}/n is the same as the action of C2subscript𝐶2C_{2} on μn=/nsubscript𝜇𝑛𝑛\mu_{n}=\mathbb{Z}/n itself, which is the sign action. The action on H4(SLn,)=c2superscriptH4subscriptSL𝑛subscript𝑐2\mathrm{H}^{4}(\operatorname{SL}_{n},\mathbb{Z})=\mathbb{Z}c_{2} is calculated by identifying H(SLn,)superscriptHsubscriptSL𝑛\mathrm{H}^{*}(\operatorname{SL}_{n},\mathbb{Z}) as a subquotient of H(BTn,)superscriptH𝐵subscript𝑇𝑛\mathrm{H}^{*}(BT_{n},\mathbb{Z}), where Tnsubscript𝑇𝑛T_{n} is the maximal torus of diagonal matrices in GLnsubscriptGL𝑛\operatorname{GL}_{n}. Specifically, H(BTn,)=[θ1,,θn]superscriptH𝐵subscript𝑇𝑛subscript𝜃1subscript𝜃𝑛\mathrm{H}^{*}(BT_{n},\mathbb{Z})=\mathbb{Z}[\theta_{1},\dots,\theta_{n}], where the C2subscript𝐶2C_{2}-action on θisubscript𝜃𝑖\theta_{i} is θiθimaps-tosubscript𝜃𝑖subscript𝜃𝑖\theta_{i}\mapsto-\theta_{i}. Then the class c2subscript𝑐2c_{2} in question may be identified with the image of the second elementary symmetric function in the θisubscript𝜃𝑖\theta_{i} in H(BTn,)/(i=1nθi)superscriptH𝐵subscript𝑇𝑛superscriptsubscript𝑖1𝑛subscript𝜃𝑖\mathrm{H}^{*}(BT_{n},\mathbb{Z})/(\sum_{i=1}^{n}\theta_{i}). It follows the action of C2subscript𝐶2C_{2} on c2subscript𝑐2c_{2} is trivial.

We know from [antieau_topological_2014, Proposition 4.4] that the illustrated d5subscript𝑑5d_{5} differential is surjective. Writing c~2subscript~𝑐2\tilde{c}_{2} for ϵnc2italic-ϵ𝑛subscript𝑐2\epsilon nc_{2}, it follows easily that the cohomology of BPGLn𝐵subscriptPGL𝑛B\operatorname{PGL}_{n} takes the stated form, and carries the stated C2subscript𝐶2C_{2}-action. ∎

Proposition 9.1.2.

Fix a natural number n𝑛n, and let ϵ=gcd(n,2)italic-ϵ𝑛2\epsilon=\gcd(n,2). Let S𝑆S be the subgroup of c2c2′′direct-sumsuperscriptsubscript𝑐2superscriptsubscript𝑐2′′c_{2}^{\prime}\mathbb{Z}\oplus c_{2}^{\prime\prime}\mathbb{Z} consisting of terms ac2+bc2′′𝑎superscriptsubscript𝑐2𝑏subscriptsuperscript𝑐′′2ac_{2}^{\prime}+bc^{\prime\prime}_{2} where a+b0(modϵn)𝑎𝑏annotated0𝑝𝑚𝑜𝑑italic-ϵ𝑛a+b\equiv 0\pmod{\epsilon n}. The low-degree cohomology of BQn𝐵subscript𝑄𝑛BQ_{n}, along with its C2subscript𝐶2C_{2}-action, is summarized by Table 2.

i𝑖i Hi(BQn,)superscriptH𝑖𝐵subscript𝑄𝑛\mathrm{H}^{i}(BQ_{n},\mathbb{Z}) Action
0 \mathbb{Z} trivial
1 0 -
2 0 -
3 α/n𝛼𝑛\alpha\mathbb{Z}/n ααmaps-to𝛼𝛼\alpha\mapsto-\alpha
4 S ac2+bc2′′bc2+ac2′′maps-to𝑎subscriptsuperscript𝑐2𝑏subscriptsuperscript𝑐′′2𝑏subscriptsuperscript𝑐2𝑎subscriptsuperscript𝑐′′2ac^{\prime}_{2}+bc^{\prime\prime}_{2}\mapsto bc^{\prime}_{2}+ac^{\prime\prime}_{2}
Table 2. The C2subscript𝐶2C_{2}-action on H(BQn,)superscriptH𝐵subscript𝑄𝑛\mathrm{H}^{*}(BQ_{n},\mathbb{Z}).

Moreover, the comparison map from Hi(BQn,)superscriptH𝑖𝐵subscript𝑄𝑛\mathrm{H}^{i}(BQ_{n},\mathbb{Z}) to Hi(BSLn,)superscriptH𝑖𝐵subscriptSL𝑛\mathrm{H}^{i}(B\operatorname{SL}_{n},\mathbb{Z}) is the evident identification map when i3𝑖3i\leq 3. When i=4𝑖4i=4, it is given by ac2+bc2′′a+bϵnc~2maps-to𝑎subscriptsuperscript𝑐2𝑏subscriptsuperscript𝑐′′2𝑎𝑏italic-ϵ𝑛subscript~𝑐2ac^{\prime}_{2}+bc^{\prime\prime}_{2}\mapsto\frac{a+b}{\epsilon n}\tilde{c}_{2}.

Proof.

There is a fibre sequence B(SLn×SLn)BQnB2μn𝐵subscriptSL𝑛subscriptSL𝑛𝐵subscript𝑄𝑛superscript𝐵2subscript𝜇𝑛B(\operatorname{SL}_{n}\times\operatorname{SL}_{n})\to BQ_{n}\to B^{2}\mu_{n}.

c2c2′′direct-sumsuperscriptsubscript𝑐2superscriptsubscript𝑐2′′\textstyle{\mathbb{Z}c_{2}^{\prime}\oplus\mathbb{Z}c_{2}^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}d5subscript𝑑5\scriptstyle{d_{5}}00\textstyle{0}00\textstyle{0}00\textstyle{0}\textstyle{\mathbb{Z}}00\textstyle{0}00\textstyle{0}α/n𝛼𝑛\textstyle{\alpha\mathbb{Z}/n}00\textstyle{0}κ/(ϵn)𝜅italic-ϵ𝑛\textstyle{\kappa\mathbb{Z}/(\epsilon n)}
Figure 3. A portion of the Serre spectral sequence in cohomology associated to B(SLn×SLn)BQnB2μn𝐵subscriptSL𝑛subscriptSL𝑛𝐵subscript𝑄𝑛superscript𝐵2subscript𝜇𝑛B(\operatorname{SL}_{n}\times\operatorname{SL}_{n})\to BQ_{n}\to B^{2}\mu_{n}.

A portion of the associated Serre spectral sequence is shown in Figure 3. There is a comparison map of spectral sequences from this one to that of Figure 2. The map identifies the bottom row of the two E2subscriptE2\mathrm{E}_{2}-pages, and sends c2,c2′′superscriptsubscript𝑐2superscriptsubscript𝑐2′′c_{2}^{\prime},c_{2}^{\prime\prime} both to c2subscript𝑐2c_{2}. It is compatible with the C2subscript𝐶2C_{2}-actions. The claimed results except the C2subscript𝐶2C_{2}-action on H4(BQn,)superscriptH4𝐵subscript𝑄𝑛\mathrm{H}^{4}(BQ_{n},\mathbb{Z}) all follow from the comparison map and the values in Table 1. As for the action on H4(BQn,)superscriptH4𝐵subscript𝑄𝑛\mathrm{H}^{4}(BQ_{n},\mathbb{Z}), as in the proof of Proposition 9.1.1, this can be deduced from the action on H(B(Tn×Tn),)=[θ1,,θn,θ1′′,,θn′′]superscriptH𝐵subscript𝑇𝑛subscript𝑇𝑛subscriptsuperscript𝜃1subscriptsuperscript𝜃𝑛subscriptsuperscript𝜃′′1subscriptsuperscript𝜃′′𝑛\mathrm{H}^{*}(B(T_{n}\times T_{n}),\mathbb{Z})=\mathbb{Z}[\theta^{\prime}_{1},\dots,\theta^{\prime}_{n},\theta^{\prime\prime}_{1},\dots,\theta^{\prime\prime}_{n}], which is given by θiθi′′maps-tosubscriptsuperscript𝜃𝑖subscriptsuperscript𝜃′′𝑖\theta^{\prime}_{i}\mapsto-\theta^{\prime\prime}_{i} and θi′′θimaps-tosubscriptsuperscript𝜃′′𝑖subscriptsuperscript𝜃𝑖\theta^{\prime\prime}_{i}\mapsto-\theta^{\prime}_{i}. ∎

Remark 9.1.3.

From Figures 2 and 3, we deduce that the maps BPGLnB2μn𝐵subscriptPGL𝑛superscript𝐵2subscript𝜇𝑛B\operatorname{PGL}_{n}\to B^{2}\mu_{n} and BQnB2μn𝐵subscript𝑄𝑛superscript𝐵2subscript𝜇𝑛BQ_{n}\to B^{2}\mu_{n} both represent generators of the groups H2(BPGLn,/n)/nsuperscriptH2𝐵subscriptPGL𝑛𝑛𝑛\mathrm{H}^{2}(B\operatorname{PGL}_{n},\mathbb{Z}/n)\cong\mathbb{Z}/n and H2(BQn,/n)/nsuperscriptH2𝐵subscript𝑄𝑛𝑛𝑛\mathrm{H}^{2}(BQ_{n},\mathbb{Z}/n)\cong\mathbb{Z}/n, respectively. Moreover, the image of the former class under the Bockstein map is a generator of H3(BPGLn,)superscriptH3𝐵subscriptPGL𝑛\mathrm{H}^{3}(B\operatorname{PGL}_{n},\mathbb{Z}), which is nothing but the tautological Brauer class α𝛼\alpha of BPGLn𝐵subscriptPGL𝑛B\operatorname{PGL}_{n}. That is, if r:XBPGLn:𝑟𝑋𝐵subscriptPGL𝑛r:X\to B\operatorname{PGL}_{n} is the classifying map for a PGLnsubscriptPGL𝑛\operatorname{PGL}_{n}-bundle, or equivalently, a degree-n𝑛n topological Azumaya algebra, then the Brauer class of that algebra is r(α)superscript𝑟𝛼r^{*}(\alpha).

Our purpose in introducing the group Qnsubscript𝑄𝑛Q_{n} is to construct a group which is as close to PGLn×PGLnsubscriptPGL𝑛subscriptPGL𝑛\operatorname{PGL}_{n}\times\operatorname{PGL}_{n} (with the interchange action) as possible, but for which the transfer of all classes in H2(BQn,/n)superscriptH2𝐵subscript𝑄𝑛𝑛\mathrm{H}^{2}(BQ_{n},\mathbb{Z}/n) vanish.

Proposition 9.1.4.

Let Qnsubscript𝑄𝑛Q_{n} be as constructed above. Give the space BQn×EC2𝐵subscript𝑄𝑛𝐸subscript𝐶2BQ_{n}\times EC_{2} the diagonal C2subscript𝐶2C_{2}-action. Then the transfer map, H2(BQn×EC2,/n)H2(BQn×C2EC2,/n)superscriptH2𝐵subscript𝑄𝑛𝐸subscript𝐶2𝑛superscriptH2superscriptsubscript𝐶2𝐵subscript𝑄𝑛𝐸subscript𝐶2𝑛\mathrm{H}^{2}(BQ_{n}\times EC_{2},\mathbb{Z}/n)\to\mathrm{H}^{2}(BQ_{n}\times^{C_{2}}EC_{2},\mathbb{Z}/n), considered at the end of Subsection 8.5, vanishes.

The space BQn×EC2𝐵subscript𝑄𝑛𝐸subscript𝐶2BQ_{n}\times EC_{2} is weakly equivalent to BQn𝐵subscript𝑄𝑛BQ_{n}, but carries a free C2subscript𝐶2C_{2}-action.

Proof.

The action of C2subscript𝐶2C_{2} on BQn×EC2𝐵subscript𝑄𝑛𝐸subscript𝐶2BQ_{n}\times EC_{2} is free. As noted in Remark 9.1.3, one generator α𝛼\alpha of H2(Qn×EC2;/n)H2(Qn;/n)superscriptH2subscript𝑄𝑛𝐸subscript𝐶2𝑛superscriptH2subscript𝑄𝑛𝑛\mathrm{H}^{2}(Q_{n}\times EC_{2};\mathbb{Z}/n)\cong\mathrm{H}^{2}(Q_{n};\mathbb{Z}/n) is given by the map BQnB2μn𝐵subscript𝑄𝑛superscript𝐵2subscript𝜇𝑛BQ_{n}\to B^{2}\mu_{n} arising from the short exact sequence defining Qnsubscript𝑄𝑛Q_{n}. This map is C2subscript𝐶2C_{2}-equivariant when μnsubscript𝜇𝑛\mu_{n}, and therefore B2μnsuperscript𝐵2subscript𝜇𝑛B^{2}\mu_{n}, is given the standard involution, and therefore the result follows from Proposition 8.5.2. ∎

Proposition 9.1.5.

Let n𝑛n be an even integer, and let a𝑎a be an odd integer. Suppose f:XBQn:𝑓𝑋𝐵subscript𝑄𝑛f:X\to BQ_{n} is a C2subscript𝐶2C_{2}-equivariant map and a 666-equivalence. Then there is no C2subscript𝐶2C_{2}-equivariant map g:XBPGLan:𝑔𝑋𝐵subscriptPGL𝑎𝑛g:X\to B\operatorname{PGL}_{an} inducing a surjection on H3(,)superscriptH3\mathrm{H}^{3}(\,\cdot\,,\mathbb{Z}).

Proof.

For the sake of contradiction, suppose that g𝑔g exists.

By Remark 9.1.3, the comoposition

XBQnB2μnB2×=B3,𝑋𝐵subscript𝑄𝑛superscript𝐵2subscript𝜇𝑛superscript𝐵2superscriptsuperscript𝐵3X\to BQ_{n}\to B^{2}\mu_{n}\to B^{2}{\mathbb{C}^{\times}}=B^{3}\mathbb{Z},

induced by f𝑓f and the inclusion μn×subscript𝜇𝑛superscript\mu_{n}\to{\mathbb{C}^{\times}}, represents a generator ξ𝜉\xi of H3(X,)=H3(BQn,)=/nsuperscriptH3𝑋superscriptH3𝐵subscript𝑄𝑛𝑛\mathrm{H}^{3}(X,\mathbb{Z})=\mathrm{H}^{3}(BQ_{n},\mathbb{Z})=\mathbb{Z}/n. As a result, there is t𝑡t\in\mathbb{Z} such that the composition

X𝑔BPGLanB2μanB2×=B3𝑔𝑋𝐵subscriptPGL𝑎𝑛superscript𝐵2subscript𝜇𝑎𝑛superscript𝐵2superscriptsuperscript𝐵3X\xrightarrow{g}B\operatorname{PGL}_{an}\to B^{2}\mu_{an}\to B^{2}{\mathbb{C}^{\times}}=B^{3}\mathbb{Z}

represents tξ𝑡𝜉t\cdot\xi. Consequently, the map XBQnB2μn𝑋𝐵subscript𝑄𝑛superscript𝐵2subscript𝜇𝑛X\to BQ_{n}\to B^{2}\mu_{n} fits into a homotopy-commutative square

X𝑋\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}g𝑔\scriptstyle{g}B2μnsuperscript𝐵2subscript𝜇𝑛\textstyle{B^{2}\mu_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}s𝑠\scriptstyle{s}BPGLan𝐵subscriptPGL𝑎𝑛\textstyle{B\operatorname{PGL}_{an}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}B2μansuperscript𝐵2subscript𝜇𝑎𝑛\textstyle{B^{2}\mu_{an}}

in which s𝑠s is the composition of B2μnB2μansuperscript𝐵2subscript𝜇𝑛superscript𝐵2subscript𝜇𝑎𝑛B^{2}\mu_{n}\to B^{2}\mu_{an} and the map B2μanB2μansuperscript𝐵2subscript𝜇𝑎𝑛superscript𝐵2subscript𝜇𝑎𝑛B^{2}\mu_{an}\to B^{2}\mu_{an} induced by xxt:μanμan:maps-to𝑥superscript𝑥𝑡subscript𝜇𝑎𝑛subscript𝜇𝑎𝑛x\mapsto x^{t}:\mu_{an}\to\mu_{an}. We extend this square into a homotopy commutative diagram

F𝐹\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}X𝑋\textstyle{X\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}g𝑔\scriptstyle{g}B2μnsuperscript𝐵2subscript𝜇𝑛\textstyle{B^{2}\mu_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}s𝑠\scriptstyle{s}BSLan𝐵subscriptSL𝑎𝑛\textstyle{B\operatorname{SL}_{an}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}BPGLan𝐵subscriptPGL𝑎𝑛\textstyle{B\operatorname{PGL}_{an}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}B2μansuperscript𝐵2subscript𝜇𝑎𝑛\textstyle{B^{2}\mu_{an}}

where both rows are homotopy fibre sequences, so F𝐹F is the homotopy fibre of XB2μn𝑋superscript𝐵2subscript𝜇𝑛X\to B^{2}\mu_{n}. Strictly speaking, we carry this out in the (fine) C2subscript𝐶2C_{2}-equivariant model structure on topological spaces, using the dual of [hovey_model_1999, Prop. 6.3.5] to deduce the existence of the dashed arrow in that category, so that it may be assumed to be C2subscript𝐶2C_{2}-equivariant. Moreover, the space F𝐹F appearing in this argument has the appropriate non-equivariant homotopy type, since the functor forgetting the C2subscript𝐶2C_{2}-action is a right Quillen functor, and therefore preserves fibre sequences.

Each of the two fibre sequences is associated to a Serre spectral sequence in cohomology. In the case of the lower row, the E2subscriptE2\mathrm{E}_{2}-page is represented in Figure 2, whereas in the case of the upper row, since X𝑋X is 666-equivalent to BQn𝐵subscript𝑄𝑛BQ_{n}, it is isomorphic on the E2subscriptE2\mathrm{E}_{2}-page to the spectral sequence represented in Figure 3. There is an induced map between these spectral sequences, and this map restricts to the following on the E2,0superscriptsubscriptE20\mathrm{E}_{2}^{\ast,0}-line:

00/an0κ/(2an)00/n0κ/(2n).00𝑎𝑛0𝜅2𝑎𝑛00𝑛0𝜅2𝑛\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 6.05556pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&&&&&\\&&&&&&\crcr}}}\ignorespaces{\hbox{\kern-6.05556pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern-1.0pt\raise 0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\kern 30.05556pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{0}$}}}}}}}{\hbox{\kern 65.05556pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{0}$}}}}}}}{\hbox{\kern 100.05556pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\mathbb{Z}/{an}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern 115.36632pt\raise-29.5pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\lower-3.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 154.67708pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{0}$}}}}}}}{\hbox{\kern 189.67708pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\kappa\mathbb{Z}/(2an)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\kern 214.25754pt\raise-29.5pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\lower-3.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern 262.838pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\dots}$}}}}}}}{\hbox{\kern-6.05556pt\raise-40.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\mathbb{Z}}$}}}}}}}{\hbox{\kern 30.05556pt\raise-40.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{0}$}}}}}}}{\hbox{\kern 65.05556pt\raise-40.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{0}$}}}}}}}{\hbox{\kern 102.6985pt\raise-40.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\mathbb{Z}/n}$}}}}}}}{\hbox{\kern 154.67708pt\raise-40.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{0}$}}}}}}}{\hbox{\kern 192.32002pt\raise-40.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\kappa\mathbb{Z}/(2n)}$}}}}}}}{\hbox{\kern 262.838pt\raise-40.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{\dots}$}}}}}}}\ignorespaces}}}}\ignorespaces.

We know the map on H5(,)superscriptH5\mathrm{H}^{5}(\,\cdot\,,\mathbb{Z}) is surjective, because in each case, the group is generated by a class κ𝜅\kappa for which 2κ2𝜅2\kappa is β(ι2)𝛽superscript𝜄2\beta(\iota^{2}), obtained by taking the canonical class ι𝜄\iota in H2(B2μn,/n)superscriptH2superscript𝐵2subscript𝜇𝑛𝑛\mathrm{H}^{2}(B^{2}\mu_{n},\mathbb{Z}/n), resp. H2(B2μan,/an)superscriptH2superscript𝐵2subscript𝜇𝑎𝑛𝑎𝑛\mathrm{H}^{2}(B^{2}\mu_{an},\mathbb{Z}/an), squaring it, and applying the Bockstein map with image H5(,)superscriptH5\mathrm{H}^{5}(\cdot,\mathbb{Z}). This may be deduced from [cartan_determination_1954], or from the Serre spectral sequence associated to the path-loop fibration BμnB2μnB\mu_{n}\to\ast\to B^{2}\mu_{n}.

Since the map gsuperscript𝑔g^{*} of spectral sequences is compatible with the C2subscript𝐶2C_{2}-action, it induces the following commutative square

c2subscript𝑐2\textstyle{c_{2}\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}d5subscript𝑑5\scriptstyle{d_{5}}gsuperscript𝑔\scriptstyle{g^{*}}κ/(2an)𝜅2𝑎𝑛\textstyle{\kappa\mathbb{Z}/(2an)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}gsuperscript𝑔\scriptstyle{g^{*}}c2c2′′direct-sumsubscriptsuperscript𝑐2subscriptsuperscript𝑐′′2\textstyle{c^{\prime}_{2}\mathbb{Z}\oplus c^{\prime\prime}_{2}\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}d5subscript𝑑5\scriptstyle{d_{5}}κ/(2n).𝜅2𝑛\textstyle{\kappa\mathbb{Z}/(2n).}

in which all arrows are C2subscript𝐶2C_{2}-equivariant. Furthermore, the proofs of Propositions 9.1.1 and 9.1.2 imply that both horizontal maps are surjective, that C2subscript𝐶2C_{2} acts trivially on c2subscript𝑐2c_{2}\mathbb{Z}, κ/(2an)𝜅2𝑎𝑛\kappa\mathbb{Z}/(2an) and κ/(2n)𝜅2𝑛\kappa\mathbb{Z}/(2n), and that the non-trivial element of C2subscript𝐶2C_{2} interchanges c2subscriptsuperscript𝑐2c^{\prime}_{2} and c2′′subscriptsuperscript𝑐′′2c^{\prime\prime}_{2}. Now, g(c2)superscript𝑔subscript𝑐2g^{*}(c_{2}) lies in the C2subscript𝐶2C_{2}-fixed subgroup of c2c2′′direct-sumsuperscriptsubscript𝑐2superscriptsubscript𝑐2′′c_{2}^{\prime}\mathbb{Z}\oplus c_{2}^{\prime\prime}\mathbb{Z}, which is to say g(c2)=mc2+mc2′′superscript𝑔subscript𝑐2𝑚superscriptsubscript𝑐2𝑚superscriptsubscript𝑐2′′g^{*}(c_{2})=mc_{2}^{\prime}+mc_{2}^{\prime\prime} for some integer m𝑚m. Then d5(g(c2))subscript𝑑5superscript𝑔subscript𝑐2d_{5}(g^{*}(c_{2})) is 2m2𝑚2m times a generator of κ/(2n)𝜅2𝑛\kappa\mathbb{Z}/(2n), and hence not a generator of κ/(2n)𝜅2𝑛\kappa\mathbb{Z}/(2n). On the other hand, g(d5(c2))superscript𝑔subscript𝑑5subscript𝑐2g^{*}(d_{5}(c_{2})) is a generator of /(2n)2𝑛\mathbb{Z}/(2n) by the previous paragraph, a contradiction. ∎

9.2. An Algebraic Counterexample

In this section, we consider complex algebraic varieties. In particular, all algebraic groups are complex algebraic groups. Cohomology is understood to be étale cohomology in the context of varieties and singular cohomology in the context of topological spaces.

A C2subscript𝐶2C_{2}-action on a variety X𝑋X will be called free if there exists a C2subscript𝐶2C_{2}-torsor XY𝑋𝑌X\to Y. In this case, Y𝑌Y coincides with categorical quotient X/C2𝑋subscript𝐶2X/C_{2} in the category of varieties. Furthermore, if C2subscript𝐶2C_{2} acts freely on X𝑋X, then it also acts freely on X()𝑋X(\mathbb{C}). The converse holds when X𝑋X is affine or projective, see Example 4.3.3 and Proposition 4.5.3, but not in general.

Fix an even positive integer n𝑛n. We define the complex algebraic group Qnsubscript𝑄𝑛Q_{n} by means of the short exact sequence

1μnx(x,x)SLn×SLnQn11subscript𝜇𝑛maps-to𝑥𝑥𝑥subscriptSL𝑛subscriptSL𝑛subscript𝑄𝑛11\to\mu_{n}\xrightarrow{x\mapsto(x,x)}\operatorname{SL}_{n}\times\operatorname{SL}_{n}\to Q_{n}\to 1

so that Qn()subscript𝑄𝑛Q_{n}(\mathbb{C}) is the group Qnsubscript𝑄𝑛Q_{n} considered in the previous subsection. There is a natural map H1(,Qn)H2(,μn)superscriptH1subscript𝑄𝑛superscriptH2subscript𝜇𝑛\mathrm{H}^{1}(\,\cdot\,,Q_{n})\to\mathrm{H}^{2}(\,\cdot\,,\mu_{n}). Composing with the map H2(,μn)H2(,𝔾m)superscriptH2subscript𝜇𝑛superscriptH2subscript𝔾𝑚\mathrm{H}^{2}(\,\cdot\,,\mu_{n})\to\mathrm{H}^{2}(\,\cdot\,,\mathbb{G}_{m}) induced by the inclusion μn𝔾msubscript𝜇𝑛subscript𝔾𝑚\mu_{n}\to\mathbb{G}_{m} allows us to associate with every Qnsubscript𝑄𝑛Q_{n}-torsor PX𝑃𝑋P\to X an n𝑛n-torsion class in H2(X,𝔾m)superscriptH2𝑋subscript𝔾𝑚\mathrm{H}^{2}(X,\mathbb{G}_{m}). This association is natural, and is, in particular, compatible with complex realization.

The first projection π1:SLn×SLnSLn:subscript𝜋1subscriptSL𝑛subscriptSL𝑛subscriptSL𝑛\pi_{1}:\operatorname{SL}_{n}\times\operatorname{SL}_{n}\to\operatorname{SL}_{n} induces a group homomorphism π1:QnPGLn:subscript𝜋1subscript𝑄𝑛subscriptPGL𝑛\pi_{1}:Q_{n}\to\operatorname{PGL}_{n} (it is not C2subscript𝐶2C_{2}-equivariant). Using this map, we associate to every Qnsubscript𝑄𝑛Q_{n}-torsor P𝑃P a PGLnsubscriptPGL𝑛\operatorname{PGL}_{n}-torsor, namely, P×QnPGLnsuperscriptsubscript𝑄𝑛𝑃subscriptPGL𝑛P\times^{Q_{n}}\operatorname{PGL}_{n}.

Lemma 9.2.1.

With the previous notation, let PX𝑃𝑋P\to X be a Qnsubscript𝑄𝑛Q_{n}-torsor, let α𝛼\alpha be its associated class in H2(X,𝔾m)superscriptH2𝑋subscript𝔾𝑚\mathrm{H}^{2}(X,\mathbb{G}_{m}), and let SX𝑆𝑋S\to X be its associated PGLnsubscriptPGL𝑛\operatorname{PGL}_{n}-torsor. Then α𝛼\alpha is the image of S𝑆S under the canonical map H1(X,PGLn)H2(X,𝔾m)superscriptH1𝑋subscriptPGL𝑛superscriptH2𝑋subscript𝔾𝑚\mathrm{H}^{1}(X,\operatorname{PGL}_{n})\to\mathrm{H}^{2}(X,\mathbb{G}_{m}). In particular, αBr(X)𝛼Br𝑋\alpha\in\operatorname{Br}(X).

Proof.

This follows by considering the following morphism of short exact sequences and the induced morphism between the associated cohomology exact sequences.

μnsubscript𝜇𝑛\textstyle{\mu_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}SLn×SLnsubscriptSL𝑛subscriptSL𝑛\textstyle{\operatorname{SL}_{n}\times\operatorname{SL}_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}π1subscript𝜋1\scriptstyle{\pi_{1}}Qnsubscript𝑄𝑛\textstyle{Q_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}π1subscript𝜋1\scriptstyle{\pi_{1}}𝔾msubscript𝔾𝑚\textstyle{\mathbb{G}_{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}GLnsubscriptGL𝑛\textstyle{\operatorname{GL}_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}PGLnsubscriptPGL𝑛\textstyle{\operatorname{PGL}_{n}}

Note that the vertical maps are not necessarily C2subscript𝐶2C_{2}-equivariant. ∎

Proposition 9.2.2.

Maintaining the previous notation, there exists a smooth affine complex variety X𝑋X with free C2subscript𝐶2C_{2}-action, a Qnsubscript𝑄𝑛Q_{n}-torsor PX𝑃𝑋P\to X and a map f:X()BQn():𝑓𝑋𝐵subscript𝑄𝑛f:X(\mathbb{C})\to BQ_{n}(\mathbb{C}) such that the following hold:

  1. (i)

    The map f:X()BQn():𝑓𝑋𝐵subscript𝑄𝑛f:X(\mathbb{C})\to BQ_{n}(\mathbb{C}) is C2subscript𝐶2C_{2}-equivariant and a 666-equivalence.

  2. (ii)

    The homotopy class of f𝑓f corresponds to the principal Qn()subscript𝑄𝑛Q_{n}(\mathbb{C})-bundle P()X()𝑃𝑋P(\mathbb{C})\to X(\mathbb{C}).

  3. (iii)

    The Brauer class αBr(X)H2(X,𝔾m)𝛼Br𝑋superscriptH2𝑋subscript𝔾𝑚\alpha\in\operatorname{Br}(X)\subseteq\mathrm{H}^{2}(X,\mathbb{G}_{m}) associated with PX𝑃𝑋P\to X has trivial image under transf:H2(X,𝔾m)H2(X/C2,𝔾m):transfsuperscriptH2𝑋subscript𝔾𝑚superscriptH2𝑋subscript𝐶2subscript𝔾𝑚\operatorname{transf}:\mathrm{H}^{2}(X,\mathbb{G}_{m})\to\mathrm{H}^{2}(X/C_{2},\mathbb{G}_{m}).

For later reference, and in keeping with the previous parts of this paper, we denote X/C2𝑋subscript𝐶2X/C_{2} by Y𝑌Y.

Proof.

As in Subsection 9.1, the group Qnsubscript𝑄𝑛Q_{n} is an affine algebraic group equipped with an algebraic C2subscript𝐶2C_{2}-action. Consequently, the split exact extension ΓΓ\Gamma in

1QnΓC211subscript𝑄𝑛Γsubscript𝐶211\to Q_{n}\to\Gamma\to C_{2}\to 1

is also an affine algebraic group, [molnar_semi-direct_1977]*Ex. 2.15 (c).

Therefore it is possible to follow [totaro_chow_1999] and construct affine spaces V𝑉V on which ΓΓ\Gamma acts and such that V𝑉V becomes a ΓΓ\Gamma-torsor after removing a locus, Z𝑍Z, of arbitrarily large codimension. In particular, VZ𝑉𝑍V-Z is a Qnsubscript𝑄𝑛Q_{n}-torsor. Choose V𝑉V so that Z𝑍Z has (complex) codimension at least 444.

Let P=(VZ)𝑃𝑉𝑍P=(V-Z) and X=(VZ)/Qn𝑋𝑉𝑍subscript𝑄𝑛X=(V-Z)/Q_{n}, and note that both P𝑃P and X𝑋X carry C2subscript𝐶2C_{2}-actions. The C2subscript𝐶2C_{2}-action on X𝑋X is free since VZ𝑉𝑍V-Z is a ΓΓ\Gamma-torsor. Moreover, one checks directly that P()X()𝑃𝑋P(\mathbb{C})\to X(\mathbb{C}) is a principal Qn()subscript𝑄𝑛Q_{n}(\mathbb{C})-bundle with involution, see Definition 8.2.1. Since C2subscript𝐶2C_{2} acts freely on X()𝑋X(\mathbb{C}), Proposition 8.2.5 implies that this principal bundle is represented by a map f:X()BQn():𝑓𝑋𝐵subscript𝑄𝑛f:X(\mathbb{C})\to BQ_{n}(\mathbb{C}), which satisfies conditions (i) and (ii).

By means of the equivariant Jouanolou device, [hoyois_six_2017]*Prop. 2.20, we may assume that X𝑋X is a smooth affine variety with these properties.

Let αBr(X)𝛼Br𝑋\alpha\in\operatorname{Br}(X) denote the Brauer class associated with PX𝑃𝑋P\to X. It remains to show that transfX/Y(α)=0subscripttransf𝑋𝑌𝛼0\operatorname{transf}_{X/Y}(\alpha)=0 in Br(Y)Br𝑌\operatorname{Br}(Y), where Y=X/C2𝑌𝑋subscript𝐶2Y=X/C_{2}.

To that end, let ξ𝜉\xi be the image of P𝑃P under H1(X,Q)H2(X,μn)superscriptH1𝑋𝑄superscriptH2𝑋subscript𝜇𝑛\mathrm{H}^{1}(X,Q)\to\mathrm{H}^{2}(X,\mu_{n}), and similarly define ξ()𝜉\xi(\mathbb{C}) as the image of P()𝑃P(\mathbb{C}) under the analogous map in singular cohomology. It is enough to check that transf(ξ)H2(Y,μn)transf𝜉superscriptH2𝑌subscript𝜇𝑛\operatorname{transf}(\xi)\in\mathrm{H}^{2}(Y,\mu_{n}) vanishes. There is a commutative diagram

Hét2(X,μn)superscriptsubscriptHét2𝑋subscript𝜇𝑛\textstyle{\mathrm{H}_{\text{\'{e}t}}^{2}(X,\mu_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}transftransf\scriptstyle{\operatorname{transf}}\scriptstyle{\cong}Hét(Y,μn)subscriptHét𝑌subscript𝜇𝑛\textstyle{\mathrm{H}_{\text{\'{e}t}}(Y,\mu_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}\scriptstyle{\cong}H2(X(),/n)superscriptH2𝑋𝑛\textstyle{\mathrm{H}^{2}(X(\mathbb{C}),\mathbb{Z}/n)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}H2(Y(),/n)superscriptH2𝑌𝑛\textstyle{\mathrm{H}^{2}(Y(\mathbb{C}),\mathbb{Z}/n)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Hét2(X,𝔾m)superscriptsubscriptHét2𝑋subscript𝔾𝑚\textstyle{\mathrm{H}_{\text{\'{e}t}}^{2}(X,\mathbb{G}_{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Hét2(Y,𝔾m)superscriptsubscriptHét2𝑌subscript𝔾𝑚\textstyle{\mathrm{H}_{\text{\'{e}t}}^{2}(Y,\mathbb{G}_{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}H3(X(),)superscriptH3𝑋\textstyle{\mathrm{H}^{3}(X(\mathbb{C}),\mathbb{Z})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}H3(Y(),)superscriptH3𝑌\textstyle{\mathrm{H}^{3}(Y(\mathbb{C}),\mathbb{Z})}

where each map from left to right is a transfer map, each map from back to front is a complex-realization map, and the maps from top to bottom are induced by the inclusion μn𝔾msubscript𝜇𝑛subscript𝔾𝑚\mu_{n}\to\mathbb{G}_{m}. The two indicated maps are isomorphism by Artin’s theorem. We also remark that H2(,×)H3(,)superscriptH2superscriptsuperscriptH3\mathrm{H}^{2}(\,\cdot\,,\mathbb{C}^{\times})\cong\mathrm{H}^{3}(\,\cdot\,,\mathbb{Z}), where the first group is understood as sheaf cohomology with coefficients in the sheaf of nonvanishing continuous complex-valued functions. Now, the transfer of ξ()H2(X(),/n)𝜉superscriptH2𝑋𝑛\xi(\mathbb{C})\in\mathrm{H}^{2}(X(\mathbb{C}),\mathbb{Z}/n) is easily seen to be 00 by comparison with H2(BQn×EC2,/n)superscriptH2𝐵subscript𝑄𝑛𝐸subscript𝐶2𝑛\mathrm{H}^{2}(BQ_{n}\times EC_{2},\mathbb{Z}/n), where it is known to vanish by Proposition 9.1.4. This completes the proof. ∎

Theorem 9.2.3.

For any even integer n𝑛n, there exists a quadratic étale map XY𝑋𝑌X\to Y of smooth affine complex varieties and an Azumaya algebra A𝐴A of degree n𝑛n over X𝑋X such that:

  1. (i)

    The period and index of α=[A]𝛼delimited-[]𝐴\alpha=[A] are both n𝑛n.

  2. (ii)

    transfX/Y(α)=0subscripttransf𝑋𝑌𝛼0\operatorname{transf}_{X/Y}(\alpha)=0 in Br(Y)Br𝑌\operatorname{Br}(Y).

  3. (iii)

    The degree of any Azumaya algebra Brauer equivalent to A𝐴A and admitting a λ𝜆\lambda-involution is divisible by 2n2𝑛2n—here λ𝜆\lambda denotes the non-trivial involution of X𝑋X over Y𝑌Y.

In particular, we see that the minimal degree of an Azumaya algebra Brauer equivalent to A𝐴A and supporting a λ𝜆\lambda-involution is at least 2n2𝑛2n. This bound is sharp by Theorem 6.3.3.

Proof.

Construct PX𝑃𝑋P\to X as in Proposition 9.2.2, and let A𝐴A be the Azumaya algebra corresponding to the PGLnsubscriptPGL𝑛\operatorname{PGL}_{n}-torsor associated to P𝑃P by means of π1:QnPGLn:subscript𝜋1subscript𝑄𝑛subscriptPGL𝑛\pi_{1}:Q_{n}\to\operatorname{PGL}_{n}. The Azumaya algebra A𝐴A has degree n𝑛n, and the complex reaization of its Brauer class is a generator of H3(X(),)/nsuperscriptH3𝑋𝑛\mathrm{H}^{3}(X(\mathbb{C}),\mathbb{Z})\cong\mathbb{Z}/n, since the map BQn()BPGLn()𝐵subscript𝑄𝑛𝐵subscriptPGL𝑛BQ_{n}(\mathbb{C})\to B\operatorname{PGL}_{n}(\mathbb{C}) induces an isomorphism on H3(,)superscriptH3\mathrm{H}^{3}(\cdot,\mathbb{Z}), see Remark 9.1.3 and the proof of Proposition 9.1.2. In particular, the period and index of α𝛼\alpha must both be n𝑛n.

Let a𝑎a be an odd integer and suppose A𝐴A were equivalent to an Azumaya algebra of degree an𝑎𝑛an carrying a λ𝜆\lambda-involution. Then, by Proposition 8.3.1, the complex realization of this algebra would correspond to a topological PGLan()subscriptPGL𝑎𝑛\operatorname{PGL}_{an}(\mathbb{C})-bundle with involution. By Proposition 8.2.5, there would be a C2subscript𝐶2C_{2}-equivariant map g:X()BPGLan():𝑔𝑋𝐵subscriptPGL𝑎𝑛g:X(\mathbb{C})\to B\operatorname{PGL}_{an}(\mathbb{C}) such that αBr(X())𝛼Br𝑋\alpha\in\operatorname{Br}(X(\mathbb{C})) was the image of the canonical Brauer class in H3(BPGLan(),)superscriptH3𝐵subscriptPGL𝑎𝑛\mathrm{H}^{3}(B\operatorname{PGL}_{an}(\mathbb{C}),\mathbb{Z}), and therefore g𝑔g would induce a surjection in H3(,)superscriptH3\mathrm{H}^{3}(\cdot,\mathbb{Z}), since the image of gsuperscript𝑔g^{*} would contain α𝛼\alpha. This is forbidden by Proposition 9.1.5. ∎

Question 9.2.4.

Does Theorem 9.2.3 hold when n𝑛n is odd?

Appendix A The Stalks of The Ring of Continuous Complex Functions

Let X𝑋X be a topological space; we work throughout on the small site of X𝑋X. Let 𝒪𝒪\mathcal{O} denote the sheaf of continuous \mathbb{C}-valued functions on X𝑋X.

Let p𝑝p be a point of X𝑋X and consider p𝒪superscript𝑝𝒪p^{*}\mathcal{O}. It is a local ring with maximal ideal denoted 𝔪𝔪\mathfrak{m}. An element fp𝒪𝑓superscript𝑝𝒪f\in p^{*}\mathcal{O} is the germ of a continuous \mathbb{C}-valued function at p𝑝p, and the class f¯p𝒪/𝔪¯𝑓superscript𝑝𝒪𝔪\bar{f}\in p^{*}\mathcal{O}/\mathfrak{m}\cong\mathbb{C} is the complex number f(p)𝑓𝑝f(p).

Proposition A.0.1.

The local ring p𝒪superscript𝑝𝒪p^{*}\mathcal{O} is strictly henselian.

Proof.

It suffices to prove the ring is a henselian ring as the residue field is \mathbb{C}.

Consider Rn:=nassignsubscript𝑅𝑛superscript𝑛R_{n}:=\mathbb{C}^{n} as an ordered set of roots of a degree-n𝑛n monic polynomial. There is a permutation action of the symmetric group ΣnsubscriptΣ𝑛\Sigma_{n} on Rnsubscript𝑅𝑛R_{n}, and there is a homeomorphism Rn/Σnnsubscript𝑅𝑛subscriptΣ𝑛superscript𝑛R_{n}/\Sigma_{n}\to\mathbb{C}^{n}, where the map takes (α1,,αn)subscript𝛼1subscript𝛼𝑛(\alpha_{1},\dots,\alpha_{n}) to the coefficients of the polynomial i=1n(tαi)superscriptsubscriptproduct𝑖1𝑛𝑡subscript𝛼𝑖\prod_{i=1}^{n}(t-\alpha_{i}), [bhatia_space_1983].

We embed Σn1ΣnsubscriptΣ𝑛1subscriptΣ𝑛\Sigma_{n-1}\subset\Sigma_{n} as the permutations fixing the first element, and form Rn/Σn1subscript𝑅𝑛subscriptΣ𝑛1R_{n}/\Sigma_{n-1}.

The space Rn/Σn1=×Rn1/Σn1subscript𝑅𝑛subscriptΣ𝑛1subscript𝑅𝑛1subscriptΣ𝑛1R_{n}/\Sigma_{n-1}=\mathbb{C}\times R_{n-1}/\Sigma_{n-1} represents monic polynomials of degree n𝑛n and a distinguished ‘first’ root, (tα1)q(t)𝑡subscript𝛼1𝑞𝑡(t-\alpha_{1})q(t). There is a closed subset ΓRn/Σn1Γsubscript𝑅𝑛subscriptΣ𝑛1\Gamma\subset R_{n}/\Sigma_{n-1}, the locus where q(α1)=0𝑞subscript𝛼10q(\alpha_{1})=0. Then Rn/Σn1ΓRn/Σn1subscript𝑅𝑛subscriptΣ𝑛1Γsubscript𝑅𝑛subscriptΣ𝑛1R_{n}/\Sigma_{n-1}-\Gamma\subset R_{n}/\Sigma_{n-1} is an open subset representing the set of monic, degree-n𝑛n polynomials having a distinguished ‘first’ root which is not repeated. Since quotient maps of spaces given by finite group actions are open maps, in the following diagram, every map appearing is an open map:

×Rn1/Σn1Γsubscript𝑅𝑛1subscriptΣ𝑛1Γ\textstyle{\mathbb{C}\times R_{n-1}/\Sigma_{n-1}-\Gamma\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}π𝜋\scriptstyle{\pi}×Rn1/Σn1subscript𝑅𝑛1subscriptΣ𝑛1\textstyle{\mathbb{C}\times R_{n-1}/\Sigma_{n-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Rnsubscript𝑅𝑛\textstyle{R_{n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}Rn/Σn.subscript𝑅𝑛subscriptΣ𝑛\textstyle{R_{n}/\Sigma_{n}.}

We denote the composite map ×Rn1/Σn1ΓRn/Σnsubscript𝑅𝑛1subscriptΣ𝑛1Γsubscript𝑅𝑛subscriptΣ𝑛\mathbb{C}\times R_{n-1}/\Sigma_{n-1}-\Gamma\to R_{n}/\Sigma_{n} by π𝜋\pi. It sends a pair (α1,q(t))subscript𝛼1𝑞𝑡(\alpha_{1},q(t)), for which q(α1)0𝑞subscript𝛼10q(\alpha_{1})\neq 0, to (tα1)q(t)𝑡subscript𝛼1𝑞𝑡(t-\alpha_{1})q(t).

Let (α1,q(t))×Rn1/Σn1Γsubscript𝛼1𝑞𝑡subscript𝑅𝑛1subscriptΣ𝑛1Γ(\alpha_{1},q(t))\in\mathbb{C}\times R_{n-1}/\Sigma_{n-1}-\Gamma be such a pair. We claim that there exists an open neighbourhood V𝑉V of (α1,q(t))subscript𝛼1𝑞𝑡(\alpha_{1},q(t)) such that π|V:VRn/Σn:evaluated-at𝜋𝑉𝑉subscript𝑅𝑛subscriptΣ𝑛\pi|_{V}:V\to R_{n}/\Sigma_{n} is a homeomorphism onto its image. Choose an open neighbourhood of (α1,q(t))×Rn1/Σn1subscript𝛼1𝑞𝑡subscript𝑅𝑛1subscriptΣ𝑛1(\alpha_{1},q(t))\in\mathbb{C}\times R_{n-1}/\Sigma_{n-1} of the form V=B(α1;ϵ)×B(q(t);ϵ)𝑉𝐵subscript𝛼1italic-ϵ𝐵𝑞𝑡italic-ϵV=B(\alpha_{1};\epsilon)\times B(q(t);\epsilon), being the product of an ϵitalic-ϵ\epsilon-ball around α1subscript𝛼1\alpha_{1} and around q(t)𝑞𝑡q(t), where ϵitalic-ϵ\epsilon is sufficiently small that none of the polynomials in B(q(t);ϵ)𝐵𝑞𝑡italic-ϵB(q(t);\epsilon) has any of the complex numbers in B(α1;ϵ)𝐵subscript𝛼1italic-ϵB(\alpha_{1};\epsilon) as roots. It is immediate that π𝜋\pi is injective when restricted to this open set in ×Rn1/Σn1Γsubscript𝑅𝑛1subscriptΣ𝑛1Γ\mathbb{C}\times R_{n-1}/\Sigma_{n-1}-\Gamma. Since π𝜋\pi is an open map, π|Vevaluated-at𝜋𝑉\pi|_{V} is a homeomorphism onto its image, establishing the claim.

Suppose we are given a polynomial h(t)p𝒪[t]𝑡superscript𝑝𝒪delimited-[]𝑡h(t)\in p^{*}\mathcal{O}[t]. Suppose further that a non-repeated root, α¯1subscript¯𝛼1\bar{\alpha}_{1}, of h¯(t)¯𝑡\bar{h}(t) is given, where h¯(t)¯𝑡\bar{h}(t) is the reduction of h(t)𝑡h(t) to (p𝒪/𝔪)[t]=[t]superscript𝑝𝒪𝔪delimited-[]𝑡delimited-[]𝑡(p^{*}{\mathcal{O}}/{\mathfrak{m}})[t]=\mathbb{C}[t]. We can write q¯(t)=h¯(t)/(tα¯1)¯𝑞𝑡¯𝑡𝑡subscript¯𝛼1\bar{q}(t)=\bar{h}(t)/(t-\bar{\alpha}_{1}) in [t]delimited-[]𝑡\mathbb{C}[t]. Note that we do not yet assert that q¯(t)¯𝑞𝑡\bar{q}(t) and α¯1subscript¯𝛼1\bar{\alpha}_{1} are the reductions of any specific elements in p𝒪[t]superscript𝑝𝒪delimited-[]𝑡p^{*}\mathcal{O}[t] or p𝒪superscript𝑝𝒪p^{*}\mathcal{O}. To prove that the ring is Henselian, we must find an element α1subscript𝛼1\alpha_{1} lifting α¯1subscript¯𝛼1\bar{\alpha}_{1} and satisfying h(α1)=0subscript𝛼10h(\alpha_{1})=0.

The germ h(t)𝑡h(t) has an extension to an open neighbourhood Up𝑝𝑈U\ni p.

We have the data of a diagram

p𝑝\textstyle{p\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}(α¯1,¯q)subscript¯𝛼1¯,𝑞\scriptstyle{(\bar{\alpha}_{1}\bar{,}q)}×Rn1/Σn1Γsubscript𝑅𝑛1subscriptΣ𝑛1Γ\textstyle{\mathbb{C}\times R_{n-1}/\Sigma_{n-1}-\Gamma\ignorespaces\ignorespaces\ignorespaces\ignorespaces}π𝜋\scriptstyle{\pi}U𝑈\textstyle{U\ignorespaces\ignorespaces\ignorespaces\ignorespaces}h\scriptstyle{h}R/Σn.𝑅subscriptΣ𝑛\textstyle{R/\Sigma_{n}.}

Around the image of (α¯1,q¯)subscript¯𝛼1¯𝑞(\bar{\alpha}_{1},\bar{q}) in ×Rn1/Σn1Γsubscript𝑅𝑛1subscriptΣ𝑛1Γ\mathbb{C}\times R_{n-1}/\Sigma_{n-1}-\Gamma we can find an open set V𝑉V such that π|V:VR/Σn:evaluated-at𝜋𝑉𝑉𝑅subscriptΣ𝑛\pi|_{V}:V\to R/\Sigma_{n} is a homeomorphism onto the image, W𝑊W. Then W𝑊W is an open set in R/Σn𝑅subscriptΣ𝑛R/\Sigma_{n} containing α¯1q¯=h¯subscript¯𝛼1¯𝑞¯\bar{\alpha}_{1}\bar{q}=\bar{h}. Since h(p)=h¯𝑝¯h(p)=\bar{h}, the preimage h1(W)superscript1𝑊h^{-1}(W) is an open subset of U𝑈U containing p𝑝p. Since πV:VW:subscript𝜋𝑉𝑉𝑊\pi_{V}:V\to W is a homeomorphism, we may lift the map

p𝑝\textstyle{p\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}(α1,q)subscript𝛼1𝑞\scriptstyle{(\alpha_{1},q)}×Rn1/Σn1Γsubscript𝑅𝑛1subscriptΣ𝑛1Γ\textstyle{\mathbb{C}\times R_{n-1}/\Sigma_{n-1}-\Gamma\ignorespaces\ignorespaces\ignorespaces\ignorespaces}π𝜋\scriptstyle{\pi}h1(W)superscript1𝑊\textstyle{h^{-1}(W)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}h\scriptstyle{h}R/Σn.𝑅subscriptΣ𝑛\textstyle{R/\Sigma_{n}.}

as indicated.

That is to say, there is a neighbourhood, h1(W)superscript1𝑊h^{-1}(W), of p𝑝p such that the factorization h¯(t)=(tα¯1)q¯(t)¯𝑡𝑡subscript¯𝛼1¯𝑞𝑡\bar{h}(t)=(t-\bar{\alpha}_{1})\bar{q}(t) can be extended on h1(W)superscript1𝑊h^{-1}(W) to a factorization h(t)=(tα1)q(t)𝑡𝑡subscript𝛼1𝑞𝑡h(t)=(t-\alpha_{1})q(t). In particular, the class of α1subscript𝛼1\alpha_{1} in p𝒪superscript𝑝𝒪p^{*}\mathcal{O} is a root of the polynomial h(t)𝑡h(t) extending α¯1subscript¯𝛼1\bar{\alpha}_{1}. This proves Hensel’s lemma for p𝒪superscript𝑝𝒪p^{*}\mathcal{O}. ∎

References